Difference between revisions of "Limits at Infinity and Asymptotes"

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==Formal Definition of a Limit Being Infinity==
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==Infinite Limits==
 +
Another kind of limit involves looking at what happens to <math>f(x)</math> as <math>x</math> gets very big. For example, consider the function <math>f(x)=\frac{1}{x}</math> . As <math>x</math> gets very big, <math>\frac{1}{x}</math> gets very small. In fact, <math>\frac{1}{x}</math> gets closer and closer to 0 the bigger <math>x</math> gets. Without limits it is very difficult to talk about this fact, because <math>x</math> can keep getting bigger and bigger and <math>\frac{1}{x}</math> never actually gets to 0; but the language of limits exists precisely to let us talk about the behavior of a function as it approaches something - without caring about the fact that it will never get there. In this case, however, we have the same problem as before: how big does <math>x</math> have to be to be sure that <math>f(x)</math> is really going towards 0?
  
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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
+
In this case, we want to say that, however close we want <math>f(x)</math> to get to 0, for <math>x</math> big enough <math>f(x)</math> is guaranteed to get that close. So we have yet another definition.
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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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===Definition of a limit at infinity===
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:<math>\lim_{x\to c}f(x)=\infty</math>
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: We call <math>L</math> the '''limit of <math>f(x)</math> as <math>x</math> approaches infinity''' if <math>f(x)</math> becomes '''arbitrarily close''' to <math>L</math> '''whenever''' <math>x</math> is '''sufficiently large'''.
 +
 
 +
When this holds we write
 +
:<math>\lim_{x\to\infty}f(x)=L</math>
 
or
 
or
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:<math>f(x)\to\infty</math> as <math>x\to c</math>
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:<math>f(x)\to L\quad\mbox{as}\quad x\to\infty</math>
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Similarly, we say that
+
Similarly, we call <math>L</math> the '''limit of <math>f(x)</math> as <math>x</math> approaches negative infinity''' if <math>f(x)</math> becomes '''arbitrarily close''' to <math>L</math> '''whenever''' <math>x</math> is '''sufficiently negative'''.
−
:<math>\lim_{x\to c}f(x)=-\infty</math>
+
 
−
if, for every <math>\varepsilon</math> , there exists a <math>\delta>0</math> such that for all <math>x\in D</math> with
+
When this holds we write
−
:<math>0<|x-c|<\delta</math>
+
:<math>\lim_{x\to-\infty}f(x)=L</math>
−
we have
 
−
:<math>f(x)<\varepsilon</math> .
 
−
When this holds we write  
 
−
:<math>\lim_{x\to c}f(x)=-\infty</math>
 
 
or
 
or
−
:<math>f(x)\to-\infty</math> as <math>x\to c</math> .
+
:<math>f(x)\to L\quad\mbox{as}\quad x\to-\infty</math>
 +
}}
 +
 
 +
So, in this case, we write:
 +
:<math>\quad\lim_{x\to\infty}\frac{1}{x}=0</math>
 +
and say "The limit, as <math>x</math> approaches infinity, equals <math>0</math> ," or "as <math>x</math> approaches infinity, the function approaches 0.
 +
 
 +
We can also write:
 +
:<math>\lim_{x\to-\infty}\frac{1}{x}=0</math>
 +
because making <math>x</math> very negative also forces <math>\frac{1}{x}</math> to be close to <math>0</math> .
 +
 
 +
'''Notice''', however, that infinity is not a number; it's just shorthand for saying "no matter how big." Thus, this is not the same as the regular limits we learned about in the last two chapters.
 +
 
 +
==Limits at Infinity of Rational Functions==
 +
One special case that comes up frequently is when we want to find the limit at <math>\infty</math> (or <math>-\infty</math>) of a rational function. A rational function is just one made by dividing two polynomials by each other. For example, <math>f(x)=\frac{x^3+x-6}{x^2-4x+3}</math> is a rational function. Also, any polynomial is a rational function, since <math>1</math> is just a (very simple) polynomial, so we can write the function <math>f(x)=x^2-3</math> as <math>f(x)=\frac{x^2-3}{1}</math> , the quotient of two polynomials.
 +
 
 +
Consider the numerator of a rational function as we allow the variable to grow very large (in either the positive or negative sense). The term with the highest exponent on the variable will dominate the numerator, and the other terms become more and more insignificant compared to the dominating term. The same applies to the denominator. In the limit, the other terms become negligible, and we only need to examine the dominating term in the numerator and denominator.
 +
 
 +
There is a simple rule for determining a limit of a rational function as the variable approaches infinity. Look for the term with the highest exponent on the variable in the numerator. Look for the same in the denominator. This rule is based on that information.
 +
 
 +
*If the exponent of the highest term in the numerator matches the exponent of the highest term in the denominator, the limit (at both <math>\infty</math> and <math>-\infty</math>) is the ratio of the coefficients of the highest terms.
 +
 
 +
*If the ''numerator'' has the highest term, then the fraction is called "top-heavy". If, when you divide the ''numerator'' by the ''denominator'' the resulting exponent on the variable is even, then the limit (at both <math>\infty</math> and <math>-\infty</math>) is <math>\infty</math> . If it is odd, then the limit at <math>\infty</math> is <math>\infty</math> , and the limit at <math>-\infty</math> is <math>-\infty</math> .
 +
 
 +
*If the ''denominator'' has the highest term, then the fraction is called "bottom-heavy" and the limit at both <math>\pm\infty</math> is 0.
 +
 
 +
Note that, if the numerator or denominator is a constant (including 1, as above), then this is the same as <math>x^0</math> . Also, a straight power of <math>x</math> , like <math>x^3</math> , has coefficient 1, since it is the same as <math>1x^3</math> .
 +
 
 +
===Examples===
 +
;Example 1
 +
Find <math>\lim_{x\to\infty}\frac{x-5}{x-3}</math> .
 +
 
 +
The function <math>f(x)=\frac{x-5}{x-3}</math> is the quotient of two polynomials, <math>x-5</math> and <math>x-3</math> . By our rule we look for the term with highest exponent in the numerator; it's <math>x</math> . The term with highest exponent in the denominator is also <math>x</math> . So, the limit is the ratio of their coefficients. Since <math>x=1x</math>, both coefficients are 1, <math>\lim_{x\to\infty}\frac{x-5}{x-3}=\frac11=1</math> .
 +
 
 +
;Example 2
 +
 
 +
Find <math>\lim_{x\to\infty}\frac{x^3+x-6}{x^2-4x+3}</math>.
 +
 
 +
Factoring out x^3 (the term with the highest power), we get
 +
<math> \frac{x^3(1+\frac{1}{x^2}-\frac{6}{x^3})}{x^3(\frac{1}{x}-\frac{4}{x^2}+\frac{3}{x^3})} = \frac{1+\frac{1}{x^2}-\frac{6}{x^3}}{\frac{1}{x}-\frac{4}{x^2}+\frac{3}{x^3}}</math>
 +
 
 +
We look at the terms with the highest exponents; for the numerator, it is <math>x^3</math>, while for the denominator it is <math>x^2</math>. Since the exponent on the numerator is higher, we know the limit at <math>\infty</math> will be <math>\infty</math>. We can see by factoring out and canceling the highest power that the denominator will go to 0, while the top will be a positive 1. So,
 +
:<math>\lim_{x\to\infty}\frac{x^3+x-6}{x^2-4x+3}=+\infty</math>.
  
 
==Resources==
 
==Resources==

Revision as of 17:09, 28 September 2021

Infinite Limits

Another kind of limit involves looking at what happens 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 f(x)} as gets very big. For example, consider the function . As gets very big, gets very small. In fact, gets closer and closer to 0 the bigger gets. Without limits it is very difficult to talk about this fact, because can keep getting bigger and bigger and never actually gets to 0; but the language of limits exists precisely to let us talk about the behavior of a function as it approaches something - without caring about the fact that it will never get there. In this case, however, we have the same problem as before: how big does have to be to be sure that is really going towards 0?

In this case, we want to say that, however close we want to get to 0, for big enough is guaranteed to get that close. So we have yet another definition.

Definition of a limit at infinity

We call the limit of as approaches infinity if becomes arbitrarily close to whenever is sufficiently large.

When this holds we write

or

Similarly, we call 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 negative infinity 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)} becomes arbitrarily close 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} 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} is sufficiently negative.

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\quad\mbox{as}\quad x\to-\infty}

}}

So, in this case, 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 \quad\lim_{x\to\infty}\frac{1}{x}=0}

and say "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 x} approaches infinity, equals 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} ," or "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 infinity, the function approaches 0.

We can also 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}\frac{1}{x}=0}

because 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} very negative also forces 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{1}{x}} to be close 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 0} .

Notice, however, that infinity is not a number; it's just shorthand for saying "no matter how big." Thus, this is not the same as the regular limits we learned about in the last two chapters.

Limits at Infinity of Rational Functions

One special case that comes up frequently is when we want to find the limit 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 \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 -\infty} ) of a rational function. A rational function is just one made by dividing two polynomials by each other. For example, 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)=\frac{x^3+x-6}{x^2-4x+3}} is a rational function. Also, any polynomial is a rational function, 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 1} is just a (very simple) polynomial, so we can write the function 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^2-3} 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 f(x)=\frac{x^2-3}{1}} , the quotient of two polynomials.

Consider the numerator of a rational function as we allow the variable to grow very large (in either the positive or negative sense). The term with the highest exponent on the variable will dominate the numerator, and the other terms become more and more insignificant compared to the dominating term. The same applies to the denominator. In the limit, the other terms become negligible, and we only need to examine the dominating term in the numerator and denominator.

There is a simple rule for determining a limit of a rational function as the variable approaches infinity. Look for the term with the highest exponent on the variable in the numerator. Look for the same in the denominator. This rule is based on that information.

  • If the exponent of the highest term in the numerator matches the exponent of the highest term in the denominator, the limit (at both 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} 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 -\infty} ) is the ratio of the coefficients of the highest terms.
  • If the numerator has the highest term, then the fraction is called "top-heavy". If, when you divide the numerator by the denominator the resulting exponent on the variable is even, then the limit (at both 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} 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 -\infty} ) 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 \infty} . If it is odd, then the limit 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 \infty} is , and the limit 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 -\infty} 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 -\infty} .
  • If the denominator has the highest term, then the fraction is called "bottom-heavy" and the limit at both 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 \pm\infty} is 0.

Note that, if the numerator or denominator is a constant (including 1, as above), then this is the same 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^0} . Also, a straight power 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} , like 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} , has coefficient 1, since it is the same 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 1x^3} .

Examples

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 \lim_{x\to\infty}\frac{x-5}{x-3}} .

The function 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)=\frac{x-5}{x-3}} is the quotient of two polynomials, 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-5} 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 x-3} . By our rule we look for the term with highest exponent in the numerator; it's 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} . The term with highest exponent in the denominator is also 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 limit is the ratio of their coefficients. 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=1x} , both coefficients are 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 \lim_{x\to\infty}\frac{x-5}{x-3}=\frac11=1} .

Example 2

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 \lim_{x\to\infty}\frac{x^3+x-6}{x^2-4x+3}} .

Factoring out x^3 (the term with the highest power), 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 \frac{x^3(1+\frac{1}{x^2}-\frac{6}{x^3})}{x^3(\frac{1}{x}-\frac{4}{x^2}+\frac{3}{x^3})} = \frac{1+\frac{1}{x^2}-\frac{6}{x^3}}{\frac{1}{x}-\frac{4}{x^2}+\frac{3}{x^3}}}

We look at the terms with the highest exponents; for the numerator, it 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 x^3} , while for the denominator it 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 x^2} . Since the exponent on the numerator is higher, we know the limit 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 \infty} will 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 \infty} . We can see by factoring out and canceling the highest power that the denominator will go to 0, while the top will be a positive 1. 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 \lim_{x\to\infty}\frac{x^3+x-6}{x^2-4x+3}=+\infty} .

Resources

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