Difference between revisions of "The Limit of a Function"

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[[File:Epsilon-delta.png|thumb|500px|right|Whenever a point <math>x</math> is within <math>\delta</math> units of <math>c</math> , <math>f(x)</math> is within <math>\varepsilon</math> units of <math>L</math>]]
  
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In preliminary calculus, the concept of a limit is probably the most difficult one to grasp (after all, it took mathematicians 150 years to arrive at it); it is also the most important and most useful one.
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The intuitive definition of a limit is inadequate to prove anything rigorously about it. The problem lies in the vague term "arbitrarily close". We discussed earlier that the meaning of this term is that the closer <math>x</math> gets to the specified value, the closer the function must get to the limit, so that however close we want the function to the limit, we can accomplish this by making <math>x</math> sufficiently close to our value. We can express this requirement technically as follows:
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==Formal definition of a limit==
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Let <math>f(x)</math> be a function defined on an open interval <math>D</math> that contains <math>c</math> , except possibly at <math>x=c</math> . Let <math>L</math> be a number. Then we say that
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:<math>\lim_{x\to c}f(x)=L</math>
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if, for every <math>\varepsilon>0</math> , there exists a <math>\delta>0</math> such that for all <math>x\in D</math> with
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:<math>0<|x-c|<\delta</math>
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we have
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:<math>\Big|f(x)-L\Big|<\varepsilon</math>
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To further explain, earlier we said that "however close we want the function to the limit, we can find a corresponding <math>x</math> close to our value." Using our new notation of epsilon (<math>\varepsilon</math>) and delta (<math>\delta</math>), we mean that if we want to make <math>f(x)</math> within <math>\varepsilon</math> of <math>L</math> , the limit, then we know that making <math>x</math> within <math>\delta</math> of <math>c</math> puts it there.
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Again, since this is tricky, let's resume our example from before: <math>f(x)=x^2</math> , at <math>x=2</math>. To start, let's say we want <math>f(x)</math> to be within .01 of the limit. We know by now that the limit should be 4, so we say: for <math>\varepsilon=0.01</math> , there is some <math>\delta</math> so that as long as <math>0<|x-c|<\delta</math> , then <math>\Big|f(x)-L\Big|<\varepsilon</math> .
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To show this, we can pick ''any'' <math>\delta</math> that is bigger than 0, so long as it works. For example, you might pick <math>10^{-14}</math> , because you are absolutely sure that if <math>x</math> is within <math>10^{-14}</math> of 2, then <math>f(x)</math> will be within <math>0.01</math> of 4. This <math>\delta</math> works for <math>\varepsilon=0.01</math> . But we can't just pick a specific value for <math>\varepsilon</math> , like 0.01, because we said in our definition "for '''every''' <math>\varepsilon>0</math> ." This means that we need to be able to show an infinite number of <math>\delta</math>s, one for each <math>\varepsilon</math> . We can't list an infinite number of <math>\delta</math>s!
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Of course, we know of a very good way to do this; we simply create a function, so that for every <math>\varepsilon</math> , it can give us a <math>\delta</math> . In this case, one definition of <math>\delta</math> that works is
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<math>\delta(\varepsilon)=\left\{\begin{matrix}2\sqrt2-2&\mbox{if }\epsilon\ge4\\\sqrt{\epsilon+4}-2&\mbox{if }\epsilon<4\end{matrix}\right.</math>
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So, in general, how do you show that <math>f(x)</math> tends to <math>L</math> as <math>x</math> tends to <math>c</math>? Well imagine somebody gave you a small number <math>\varepsilon</math> (e.g., say <math>\varepsilon=0.03</math>). Then you have to find a <math>\delta>0</math> and show that whenever <math>0<|x-c|<\delta</math> we have <math>\Big|f(x)-L\Big|<0.03</math> . Now if that person gave you a smaller <math>\varepsilon</math> (say <math>\varepsilon=0.002</math>) then you would have to find another <math>\delta</math>, but this time with 0.03 replaced by 0.002. If you can do this for ''any'' choice of <math>\varepsilon</math> then you have shown that <math>f(x)</math> tends to <math>L</math> as <math>x</math> tends to <math>c</math> . Of course, the way you would do this in general would be to create a function giving you a <math>\delta</math> for every <math>\varepsilon</math> , just as in the example above.
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===Formal Definition of the Limit at Infinity===
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We call <math>L</math> the '''limit''' of <math>f(x)</math> as <math>x</math> approaches <math>\infty</math> if for every number <math>\varepsilon>0</math> there exists a <math>\delta</math> such that
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whenever
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<math>x>\delta</math>
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we have
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:<math>\Big|f(x)-L\Big|<\varepsilon</math>
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When this holds we write
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:<math>\lim_{x\to\infty}f(x)=L</math>
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or
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:<math>f(x)\to L</math> as <math>x\to\infty</math>
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Similarly, we call <math>L</math> the '''limit''' of <math>f(x)</math> as <math>x</math> approaches <math>-\infty</math> if for every number <math>\varepsilon>0</math> , there exists a number <math>\delta</math> such that whenever <math>x<\delta</math> we have
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:<math>\Big|f(x)-L\Big|<\varepsilon</math>
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When this holds we write
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:<math>\lim_{x\to-\infty}f(x)=L</math>
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or
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:<math>f(x)\to L</math> as <math>x\to-\infty</math>
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Notice the difference in these two definitions. For the limit of <math>f(x)</math> as <math>x</math> approaches <math>\infty</math> we are interested in those <math>x</math> such that <math>x>\delta</math> . For the limit of <math>f(x)</math> as <math>x</math> approaches <math>-\infty</math> we are interested in those <math>x</math> such that <math>x<\delta</math> .
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==Examples==
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Here are some examples of the formal definition.
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;Example 1
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We know from earlier in the chapter that
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:<math>\lim_{x\to8}\frac{x}{4}=2</math>
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What is <math>\delta</math> when <math>\varepsilon=0.01</math> for this limit?
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We start with the desired conclusion and substitute the given values for <math>f(x)</math> and <math>\varepsilon</math> :
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:<math>\left|\frac{x}{4}-2\right|<0.01</math>
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Then we solve the inequality for <math>x</math> :
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:<math>7.96<x<8.04</math>
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This is the same as saying
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:<math>-0.04<x-8<0.04</math>
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(We want the thing in the middle of the inequality to be <math>x-8</math> because that's where we're taking the limit.)
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We normally choose the smaller of <math>|-0.04|</math> and <math>0.04</math> for <math>\delta</math>, so <math>\delta=0.04</math> , but any smaller number will also work.
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;Example 2
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What is the limit of <math>f(x)=x+7</math> as <math>x</math> approaches 4?
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There are two steps to answering such a question; first we must determine the answer — this is where intuition and guessing is useful, as well as the informal definition of a limit — and then we must prove that the answer is right.
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In this case, 11 is the limit because we know <math>f(x)=x+7</math> is a continuous function whose domain is all real numbers. Thus, we can find the limit by just substituting 4 in for <math>x</math> , so the answer is <math>4+7=11</math> .
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We're not done, though, because we never proved any of the limit laws rigorously; we just stated them. In fact, we couldn't have proved them, because we didn't have the formal definition of the limit yet, Therefore, in order to be sure that 11 is the right answer, we need to prove that no matter what value of <math>\varepsilon</math> is given to us, we can find a value of <math>\delta</math> such that
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:<math>\Big|f(x)-11\Big|<\varepsilon</math>
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whenever
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:<math>|x-4|<\delta</math>
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For this particular problem, letting <math>\delta=\varepsilon</math> works. Now, we have to prove
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:<math>\Big|f(x)-11\Big|<\varepsilon</math>
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given that
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:<math>|x-4|<\delta=\varepsilon</math>
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Since <math>|x-4|<\varepsilon</math> , we know
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:<math>\Big|f(x)-11\Big|=\Big|x+7-11\Big|=|x-4|<\varepsilon</math>
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which is what we wished to prove.
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;Example 3
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What is the limit of <math>f(x)=x^2</math> as <math>x</math> approaches 4?
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As before, we use what we learned earlier in this chapter to guess that the limit is <math>4^2=16</math> . Also as before, we pull out of thin air that
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:<math>\delta=\sqrt{\varepsilon+16}-4</math>
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Note that, since <math>\varepsilon</math> is always positive, so is <math>\delta</math> , as required. Now, we have to prove
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:<math>\Big|x^2-16\Big|<\varepsilon</math>
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given that
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:<math>|x-4|<\delta=\sqrt{\varepsilon+16}-4</math> .
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We know that
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:<math>|x+4|=\Big|(x-4)+8\Big|\le|x-4|+8<\delta+8</math>
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(because of the triangle inequality), so
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:<math>\begin{matrix}
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\Big|x^2-16\Big|&=&|x-4|\cdot|x+4|\\  \\
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\ &<&\delta(\delta+8)\\  \\
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\ &<&(\sqrt{16+\varepsilon}-4)(\sqrt{16+\varepsilon}+4) \\  \\
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\ &<&(\sqrt{16+\varepsilon})^2-4^2\\  \\
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\ &=&\varepsilon+16-16\\ \\
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\ &<&\varepsilon\end{matrix}</math>
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;Example 4
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Show that the limit of <math>\sin\left(\tfrac1x\right)</math> as <math>x</math> approaches 0 does not exist.
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We will proceed by contradiction. Suppose the limit exists; call it <math>L</math> . For simplicity, we'll assume that <math>L\ne1</math> ; the case for <math>L=1</math> is similar. Choose <math>\varepsilon=|1-L|</math> . Then if the limit were <math>L</math> there would be some <math>\delta>0</math> such that <math>\left|\sin\left(\tfrac1x\right)-L\right|<\varepsilon=|1-L|</math> for every <math>x</math> with <math>0<|x|<\delta</math> . But, for every <math>\delta>0</math> , there exists some (possibly very large) <math>n</math> such that <math>0<x_0=\frac{1}{\frac{\pi}{2}+2\pi n}<\delta</math> , but <math>\left|\sin\left(\tfrac{1}{x_0}\right)-L\right|=|1-L|</math> , a contradiction.
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;Example 5
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What is the limit of <math>x\sin\left(\tfrac1x\right)</math> as <math>x</math> approaches 0?
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By the Squeeze Theorem, we know the answer should be 0. To prove this, we let <math>\delta=\varepsilon</math> . Then for all <math>x</math> , if <math>0<|x|<\delta</math> , then <math>\left|x\sin\left(\tfrac1x\right)-0\right|\le|x|<\varepsilon</math> as required.
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;Example 6
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Suppose that <math>\lim_{x\to a}f(x)=L</math> and <math>\lim_{x\to a}g(x)=M</math> . What is <math>\lim_{x\to a}\big[f(x)+g(x)\big]</math> ?
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Of course, we know the answer should be <math>L+M</math> , but now we can prove this rigorously. Given some <math>\varepsilon</math> , we know there's a <math>\delta_1</math> such that, for any <math>x</math> with <math>0<|x-a|<\delta_1</math> , <math>\Big|f(x)-L\Big|<\frac{\varepsilon}{2}</math> (since the definition of limit says "for any <math>\varepsilon</math>", so it must be true for <math>\frac{\varepsilon}{2}</math> as well). Similarly, there's a <math>\delta_2</math> such that, for any <math>x</math> with <math>0<|x-a|<\delta_2</math> , <math>\Big|g(x)-M\Big|<\frac{\varepsilon}{2}</math> . We can set <math>\delta</math> to be the lesser of <math>\delta_1</math> and <math>\delta_2</math> . Then, for any <math>x</math> with <math>0<|x-a|<\delta</math> , <math>\bigg|\big(f(x)+g(x)\big)-(L+M)\bigg|\le\Big|f(x)-L\Big|+\Big|g(x)-M\Big|<\frac{\varepsilon}{2}+\frac{\varepsilon}{2}
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=\varepsilon</math> , as required.
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If you like, you can prove the other limit rules too using the new definition. Mathematicians have already done this, which is how we know the rules work. Therefore, when computing a limit from now on, we can go back to just using the rules and still be confident that our limit is correct according to the rigorous definition.
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<!-- Can you prove that ln(lim(x->a)f(x))=lim(x->a)ln(f(x))? -->
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==Formal Definition of a Limit Being Infinity==
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Let <math>f(x)</math> be a function defined on an open interval <math>D</math> that contains <math>c</math> , except possibly at <math>x=c</math> . Then we say that
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:<math>\lim_{x\to c}f(x)=\infty</math>
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if, for every <math>\varepsilon</math> , there exists a <math>\delta>0</math> such that for all <math>x\in D</math> with
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:<math>0<|x-c|<\delta</math>
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we have
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:<math>f(x)>\varepsilon</math> .
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When this holds we write
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:<math>\lim_{x\to c}f(x)=\infty</math>
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or
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:<math>f(x)\to\infty</math> as <math>x\to c</math>
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Similarly, we say that
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:<math>\lim_{x\to c}f(x)=-\infty</math>
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if, for every <math>\varepsilon</math> , there exists a <math>\delta>0</math> such that for all <math>x\in D</math> with
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:<math>0<|x-c|<\delta</math>
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we have
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:<math>f(x)<\varepsilon</math> .
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When this holds we write
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:<math>\lim_{x\to c}f(x)=-\infty</math>
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or
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:<math>f(x)\to-\infty</math> as <math>x\to c</math>.
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==Exact Integrals as Limits of Sums==
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Using the definition of an integral, we can evaluate the limit as <math>n</math> goes to infinity. This technique requires a fairly high degree of familiarity with summation identities. This technique is often referred to as evaluation "by definition," and can be used to find definite integrals, as long as the integrands are fairly simple. We start with definition of the integral:
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:{|
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|<math>\int\limits_a^b f(x)dx</math>
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|<math>=\lim_{n\to\infty}\left[\frac{b-a}{n}\sum_{k=1}^n f(x_k^*)\right]</math>
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|Then picking <math>x_k^*</math> to be <math>x_k=a+k\frac{b-a}{n}</math> we get,
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|-
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|
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|<math>=\lim_{n\to\infty}\left[\frac{b-a}{n}\sum_{k=1}^n f\big(a+k\tfrac{b-a}{n}\big)\right]</math>
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|}
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In some simple cases, this expression can be reduced to a real number, which can be interpreted as the area under the curve if <math>f(x)</math> is positive on <math>[a,b]</math> .
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===Example 1===
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Find <math>\int\limits_0^2x^2dx</math> by writing the integral as a limit of Riemann sums.
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:{|
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|<math>\int\limits_0^2x^2dx</math>
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|<math>=\lim_{n\to\infty}\left[\frac{b-a}{n}\sum_{k=1}^n f(x_k^*)\right]</math>
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|-
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|
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|<math>=\lim_{n\to\infty}\left[\frac{2}{n}\sum_{k=1}^n f\big(\tfrac{2k}{n}\big)\right]</math>
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|-
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|
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|<math>=\lim_{n\to\infty}\left[\frac{2}{n}\sum_{k=1}^n\left(\frac{2k}{n}\right)^2\right]</math>
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|-
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|
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|<math>=\lim_{n\to\infty}\left[\frac{2}{n}\sum_{k=1}^n\frac{4k^2}{n^2}\right]</math>
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|-
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|
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|<math>=\lim_{n\to\infty}\left[\frac{8}{n^3}\sum_{k=1}^nk^2\right]</math>
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|-
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|
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|<math>=\lim_{n\to\infty}\left[\frac{8}{n^3}\cdot\frac{n(n+1)(2n+1)}{6}\right]</math>
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|-
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|
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|<math>=\lim_{n\to\infty}\left[\frac{4}{3}\cdot\frac{2n^2+3n+1}{n^2}\right]</math>
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|-
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|
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|<math>=\lim_{n\to\infty}\left[\frac{8}{3}+\frac{4}{n}+\frac{4}{3n^2}\right]</math>
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|-
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|
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|<math>=\frac{8}{3}</math>
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|}
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In other cases, it is even possible to evaluate indefinite integrals using the formal definition. We can define the indefinite integral as follows:
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:{|
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|<math>\int f(x)dx</math>
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|<math>=\int\limits_0^x f(t)dt=\lim_{n\to\infty}\left[\frac{x-0}{n}\sum_{k=1}^n f(t_k^*)\right]</math>
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|-
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|
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|<math>=\lim_{n\to\infty}\left[\frac{x}{n}\sum_{k=1}^n f\big(0+\tfrac{k(x-0)}{n}\big)\right]</math>
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|-
 +
|
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|<math>=\lim_{n\to\infty}\left[\frac{x}{n}\sum_{k=1}^n f\big(\tfrac{kx}{n}\big)\right]</math>
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|}
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===Example 2===
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Suppose <math>f(x)=x^2</math> , then we can evaluate the indefinite integral as follows.
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:{|
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|<math>\int\limits_0^x f(t)dt</math>
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|<math>=\lim_{n\to\infty}\left[\frac{x}{n}\sum_{k=1}^n f\big(\tfrac{kx}{n}\big)\right]</math>
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|-
 +
|
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|<math>=\lim_{n\to\infty}\left[\frac{x}{n}\sum_{k=1}^n\left(\frac{kx}{n}\right)^2\right]</math>
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|-
 +
|
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|<math>=\lim_{n\to\infty}\left[\frac{x}{n}\sum_{k=1}^n\frac{k^2\cdot x^2}{n^2}\right]</math>
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|-
 +
|
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|<math>=\lim_{n\to\infty}\left[\frac{x^3}{n^3}\sum_{k=1}^n k^2\right]</math>
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|-
 +
|
 +
|<math>=\lim_{n\to\infty}\left[\frac{x^3}{n^3}\sum_{k=1}^n k^2\right]</math>
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|-
 +
|
 +
|<math>=\lim_{n\to\infty}\left[\frac{x^3}{n^3}\cdot\frac{n(n+1)(2n+1)}{6}\right]</math>
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|-
 +
|
 +
|<math>=\lim_{n\to\infty}\left[\frac{x^3}{n^3}\cdot\frac{2n^3+3n^2+n}{6}\right]</math>
 +
|-
 +
|
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|<math>=x^3\cdot\lim_{n\to\infty}\left[\frac{2n^3}{6n^3}+\frac{3n^2}{6n^3}+\frac{n}{6n^3}\right]</math>
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|-
 +
|
 +
|<math>=x^3\cdot\lim_{n\to\infty}\left[\frac{1}{3}+\frac{1}{2n}+\frac{1}{6n^2}\right]</math>
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|-
 +
|
 +
|<math>=x^3\cdot\left(\frac{1}{3}\right)</math>
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|-
 +
|
 +
|<math>=\frac{x^3}{3}</math>
 +
|}
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 +
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==Resources==
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 +
* [https://en.wikibooks.org/wiki/Calculus/Integration_techniques/Infinite_Sums Infinite Sums], Wikibooks: Calculus/Integration techniques
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* [https://en.wikibooks.org/wiki/Calculus/Formal_Definition_of_the_Limit Formal Definition of the Limit], Wikibooks: Calculus
  
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Limits%20of%20Functions/MAT1214_2.2TheLimitOfAFunctionPwPt.pptx The Limit of a Function] PowerPoint file created by Dr. Sara Shirinkam, UTSA.
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Limits%20of%20Functions/MAT1214_2.2TheLimitOfAFunctionPwPt.pptx The Limit of a Function] PowerPoint file created by Dr. Sara Shirinkam, UTSA.
  
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* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Limits%20of%20Functions/MAT1214-2.2TheLimitOfAFunctionNotes1.pdf The Limit of a Function] notes created by Instructor Beatty,UTSA.
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* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Limits%20of%20Functions/MAT1214-2.2TheLimitOfAFunctionNotes1.pdf The Limit of a Function] notes created by Instructor Beatty, UTSA.
  
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* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Limits%20of%20Functions/MAT1214_2.2TheLimitOfAFunctionNotes.pdf The Left and Right Hand Limit] notes created by Instructor Beatty,UTSA.
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* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Limits%20of%20Functions/MAT1214_2.2TheLimitOfAFunctionNotes.pdf The Left and Right Hand Limit] notes created by Instructor Beatty, UTSA.
  
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Limits%20of%20Functions/MAT1214-2.2TheLimitOfAFunctionWS1.pdf The Limit of a Function Worksheet 1]  
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Limits%20of%20Functions/MAT1214-2.2TheLimitOfAFunctionWS1.pdf The Limit of a Function Worksheet 1]  
  
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Limits%20of%20Functions/MAT1214_2.2TheLimitOfAFunctionWS2.pdf The Limit of a Function Worksheet 2]
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1214/Limits%20of%20Functions/MAT1214_2.2TheLimitOfAFunctionWS2.pdf The Limit of a Function Worksheet 2]
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* [https://youtu.be/QfPqRMqP5kU Calculus 1 - Introduction to Limits] by The Organic Chemistry Tutor
 
  
 
* [https://youtu.be/HYSI-AHUqRM What is a Limit? Basic Idea of Limits] by patrickJMT
 
* [https://youtu.be/HYSI-AHUqRM What is a Limit? Basic Idea of Limits] by patrickJMT
  
 
* [https://youtu.be/kG_p2vKApOE Lots of Limit Examples, Part 1] by patrickJMT
 
* [https://youtu.be/kG_p2vKApOE Lots of Limit Examples, Part 1] by patrickJMT
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 +
==Licensing==
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Content obtained and/or adapted from:
 +
* [https://en.wikibooks.org/wiki/Calculus/Integration_techniques/Infinite_Sums Infinite Sums, Wikibooks: Calculus/Integration techniques] under a CC BY-SA license
 +
* [https://en.wikibooks.org/wiki/Calculus/Formal_Definition_of_the_Limit Formal Definition of the Limit, Wikibooks: Calculus] under a CC BY-SA license

Latest revision as of 15:41, 15 January 2022

Whenever a point is within Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} units of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is within Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} units of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L}

In preliminary calculus, the concept of a limit is probably the most difficult one to grasp (after all, it took mathematicians 150 years to arrive at it); it is also the most important and most useful one.

The intuitive definition of a limit is inadequate to prove anything rigorously about it. The problem lies in the vague term "arbitrarily close". We discussed earlier that the meaning of this term is that the closer Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} gets to the specified value, the closer the function must get to the limit, so that however close we want the function to the limit, we can accomplish this by making Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} sufficiently close to our value. We can express this requirement technically as follows:

Formal definition of a limit

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} be a function defined on an open interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} that contains Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} , except possibly at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=c} . Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} be a number. Then we say that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c}f(x)=L}

if, for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon>0} , there exists a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta>0} such that for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in D} with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<|x-c|<\delta}

we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Big|f(x)-L\Big|<\varepsilon}

To further explain, earlier we said that "however close we want the function to the limit, we can find a corresponding Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} close to our value." Using our new notation of epsilon (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} ) and delta (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} ), we mean that if we want to make Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} within Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} , the limit, then we know that making Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} within Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} puts it there.

Again, since this is tricky, let's resume our example from before: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x^2} , at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=2} . To start, let's say we want Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} to be within .01 of the limit. We know by now that the limit should be 4, so we say: for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon=0.01} , there is some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} so that as long as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<|x-c|<\delta} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Big|f(x)-L\Big|<\varepsilon} .

To show this, we can pick any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} that is bigger than 0, so long as it works. For example, you might pick Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{-14}} , because you are absolutely sure that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is within Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 10^{-14}} of 2, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} will be within Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.01} of 4. This Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} works for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon=0.01} . But we can't just pick a specific value for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} , like 0.01, because we said in our definition "for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon>0} ." This means that we need to be able to show an infinite number of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} s, one for each Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} . We can't list an infinite number of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} s!

Of course, we know of a very good way to do this; we simply create a function, so that for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} , it can give us a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} . In this case, one definition of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} that works is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta(\varepsilon)=\left\{\begin{matrix}2\sqrt2-2&\mbox{if }\epsilon\ge4\\\sqrt{\epsilon+4}-2&\mbox{if }\epsilon<4\end{matrix}\right.}

So, in general, how do you show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} tends to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} tends to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} ? Well imagine somebody gave you a small number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} (e.g., say Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon=0.03} ). Then you have to find a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta>0} and show that whenever Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<|x-c|<\delta} we have Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Big|f(x)-L\Big|<0.03} . Now if that person gave you a smaller Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} (say Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon=0.002} ) then you would have to find another Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} , but this time with 0.03 replaced by 0.002. If you can do this for any choice of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} then you have shown that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} tends to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} tends to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} . Of course, the way you would do this in general would be to create a function giving you a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} , just as in the example above.

Formal Definition of the Limit at Infinity

We call Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} the limit of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} approaches Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty} if for every number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon>0} there exists a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} such that whenever Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x>\delta} we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Big|f(x)-L\Big|<\varepsilon}

When this holds we write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to\infty}f(x)=L}

or

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)\to L} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\to\infty}

Similarly, we call Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} the limit of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} approaches Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\infty} if for every number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon>0} , there exists a number Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} such that whenever Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x<\delta} we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Big|f(x)-L\Big|<\varepsilon}

When this holds we write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to-\infty}f(x)=L}

or

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)\to L} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\to-\infty}


Notice the difference in these two definitions. For the limit of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} approaches Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \infty} we are interested in those Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x>\delta} . For the limit of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} approaches Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\infty} we are interested in those Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x<\delta} .

Examples

Here are some examples of the formal definition.

Example 1

We know from earlier in the chapter that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to8}\frac{x}{4}=2}

What is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon=0.01} for this limit?

We start with the desired conclusion and substitute the given values for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon}  :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\frac{x}{4}-2\right|<0.01}

Then we solve the inequality for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x}  :

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 7.96<x<8.04}

This is the same as saying

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -0.04<x-8<0.04}

(We want the thing in the middle of the inequality to be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x-8} because that's where we're taking the limit.) We normally choose the smaller of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |-0.04|} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0.04} for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} , so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta=0.04} , but any smaller number will also work.

Example 2

What is the limit of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x+7} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} approaches 4?

There are two steps to answering such a question; first we must determine the answer — this is where intuition and guessing is useful, as well as the informal definition of a limit — and then we must prove that the answer is right.

In this case, 11 is the limit because we know Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x+7} is a continuous function whose domain is all real numbers. Thus, we can find the limit by just substituting 4 in for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , so the answer is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4+7=11} .

We're not done, though, because we never proved any of the limit laws rigorously; we just stated them. In fact, we couldn't have proved them, because we didn't have the formal definition of the limit yet, Therefore, in order to be sure that 11 is the right answer, we need to prove that no matter what value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} is given to us, we can find a value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} such that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Big|f(x)-11\Big|<\varepsilon}

whenever

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x-4|<\delta}

For this particular problem, letting Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta=\varepsilon} works. Now, we have to prove

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Big|f(x)-11\Big|<\varepsilon}

given that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x-4|<\delta=\varepsilon}

Since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x-4|<\varepsilon} , we know

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Big|f(x)-11\Big|=\Big|x+7-11\Big|=|x-4|<\varepsilon}

which is what we wished to prove.

Example 3

What is the limit of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x^2} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} approaches 4?

As before, we use what we learned earlier in this chapter to guess that the limit is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 4^2=16} . Also as before, we pull out of thin air that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta=\sqrt{\varepsilon+16}-4}

Note that, since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} is always positive, so is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} , as required. Now, we have to prove

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Big|x^2-16\Big|<\varepsilon}

given that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x-4|<\delta=\sqrt{\varepsilon+16}-4} .

We know that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x+4|=\Big|(x-4)+8\Big|\le|x-4|+8<\delta+8}

(because of the triangle inequality), so

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{matrix} \Big|x^2-16\Big|&=&|x-4|\cdot|x+4|\\ \\ \ &<&\delta(\delta+8)\\ \\ \ &<&(\sqrt{16+\varepsilon}-4)(\sqrt{16+\varepsilon}+4) \\ \\ \ &<&(\sqrt{16+\varepsilon})^2-4^2\\ \\ \ &=&\varepsilon+16-16\\ \\ \ &<&\varepsilon\end{matrix}}
Example 4

Show that the limit of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin\left(\tfrac1x\right)} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} approaches 0 does not exist.

We will proceed by contradiction. Suppose the limit exists; call it Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} . For simplicity, we'll assume that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L\ne1}  ; the case for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L=1} is similar. Choose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon=|1-L|} . Then if the limit were Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} there would be some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta>0} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\sin\left(\tfrac1x\right)-L\right|<\varepsilon=|1-L|} for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<|x|<\delta} . But, for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta>0} , there exists some (possibly very large) such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<x_0=\frac{1}{\frac{\pi}{2}+2\pi n}<\delta} , but Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\sin\left(\tfrac{1}{x_0}\right)-L\right|=|1-L|} , a contradiction.

Example 5

What is the limit of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\sin\left(\tfrac1x\right)} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} approaches 0?

By the Squeeze Theorem, we know the answer should be 0. To prove this, we let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta=\varepsilon} . Then for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<|x|<\delta} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|x\sin\left(\tfrac1x\right)-0\right|\le|x|<\varepsilon} as required.

Example 6

Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to a}f(x)=L} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to a}g(x)=M} . What is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to a}\big[f(x)+g(x)\big]}  ?

Of course, we know the answer should be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L+M} , but now we can prove this rigorously. Given some Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} , we know there's a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_1} such that, for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<|x-a|<\delta_1} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Big|f(x)-L\Big|<\frac{\varepsilon}{2}} (since the definition of limit says "for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} ", so it must be true for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{\varepsilon}{2}} as well). Similarly, there's a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_2} such that, for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<|x-a|<\delta_2} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Big|g(x)-M\Big|<\frac{\varepsilon}{2}} . We can set Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta} to be the lesser of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_1} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_2} . Then, for any Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} with Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<|x-a|<\delta} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bigg|\big(f(x)+g(x)\big)-(L+M)\bigg|\le\Big|f(x)-L\Big|+\Big|g(x)-M\Big|<\frac{\varepsilon}{2}+\frac{\varepsilon}{2} =\varepsilon} , as required.

If you like, you can prove the other limit rules too using the new definition. Mathematicians have already done this, which is how we know the rules work. Therefore, when computing a limit from now on, we can go back to just using the rules and still be confident that our limit is correct according to the rigorous definition.

Formal Definition of a Limit Being Infinity

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} be a function defined on an open interval Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D} that contains Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} , except possibly at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=c} . Then we say that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c}f(x)=\infty}

if, for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} , there exists a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta>0} such that for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in D} with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<|x-c|<\delta}

we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)>\varepsilon} .

When this holds we write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c}f(x)=\infty}

or

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)\to\infty} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\to c}

Similarly, we say that

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c}f(x)=-\infty}

if, for every Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon} , there exists a Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta>0} such that for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in D} with

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<|x-c|<\delta}

we have

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)<\varepsilon} .

When this holds we write

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x\to c}f(x)=-\infty}

or

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)\to-\infty} as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\to c} .

Exact Integrals as Limits of Sums

Using the definition of an integral, we can evaluate the limit as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} goes to infinity. This technique requires a fairly high degree of familiarity with summation identities. This technique is often referred to as evaluation "by definition," and can be used to find definite integrals, as long as the integrands are fairly simple. We start with definition of the integral:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\limits_a^b f(x)dx} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{b-a}{n}\sum_{k=1}^n f(x_k^*)\right]} Then picking Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_k^*} to be Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_k=a+k\frac{b-a}{n}} we get,
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{b-a}{n}\sum_{k=1}^n f\big(a+k\tfrac{b-a}{n}\big)\right]}

In some simple cases, this expression can be reduced to a real number, which can be interpreted as the area under the curve if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is positive on Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [a,b]} .

Example 1

Find Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\limits_0^2x^2dx} by writing the integral as a limit of Riemann sums.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\limits_0^2x^2dx} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{b-a}{n}\sum_{k=1}^n f(x_k^*)\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{2}{n}\sum_{k=1}^n f\big(\tfrac{2k}{n}\big)\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{2}{n}\sum_{k=1}^n\left(\frac{2k}{n}\right)^2\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{2}{n}\sum_{k=1}^n\frac{4k^2}{n^2}\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{8}{n^3}\sum_{k=1}^nk^2\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{8}{n^3}\cdot\frac{n(n+1)(2n+1)}{6}\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{4}{3}\cdot\frac{2n^2+3n+1}{n^2}\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{8}{3}+\frac{4}{n}+\frac{4}{3n^2}\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{8}{3}}

In other cases, it is even possible to evaluate indefinite integrals using the formal definition. We can define the indefinite integral as follows:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int f(x)dx} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\int\limits_0^x f(t)dt=\lim_{n\to\infty}\left[\frac{x-0}{n}\sum_{k=1}^n f(t_k^*)\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{x}{n}\sum_{k=1}^n f\big(0+\tfrac{k(x-0)}{n}\big)\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{x}{n}\sum_{k=1}^n f\big(\tfrac{kx}{n}\big)\right]}

Example 2

Suppose Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)=x^2} , then we can evaluate the indefinite integral as follows.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \int\limits_0^x f(t)dt} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{x}{n}\sum_{k=1}^n f\big(\tfrac{kx}{n}\big)\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{x}{n}\sum_{k=1}^n\left(\frac{kx}{n}\right)^2\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{x}{n}\sum_{k=1}^n\frac{k^2\cdot x^2}{n^2}\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{x^3}{n^3}\sum_{k=1}^n k^2\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{x^3}{n^3}\sum_{k=1}^n k^2\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{x^3}{n^3}\cdot\frac{n(n+1)(2n+1)}{6}\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\lim_{n\to\infty}\left[\frac{x^3}{n^3}\cdot\frac{2n^3+3n^2+n}{6}\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =x^3\cdot\lim_{n\to\infty}\left[\frac{2n^3}{6n^3}+\frac{3n^2}{6n^3}+\frac{n}{6n^3}\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =x^3\cdot\lim_{n\to\infty}\left[\frac{1}{3}+\frac{1}{2n}+\frac{1}{6n^2}\right]}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle =\frac{x^3}{3}}


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