Difference between revisions of "Real Function Limits:Sequential Criterion"

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<h1 id="toc0"><span>The Sequential Criterion for a Limit of a Function</span></h1>
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<p>We will now look at a very important theorem known as The Sequential Criterion for a Limit which merges the concept of the limit of a function <math>f</math> at a cluster point <math>c/math> from <math>A</math> with regards to sequences <math>(a_n)</math> from <math>A</math> that converge to <math>c</math>.</p>
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<p>We will now look at a very important theorem known as '''The Sequential Criterion for a Limit''' which merges the concept of the limit of a function <math>f</math> at a cluster point <math>c</math> from <math>A</math> with regards to sequences <math>(a_n)</math> from <math>A</math> that converge to <math>c</math>.</p>
<table class="wiki-content-table">
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<tr>
 
<tr>
<td><strong>Theorem 1 (The Sequential Criterion for a Limit of a Function):</strong> Let <span class="math-inline"><math>f : A \to \mathbb{R}</math></span> be a function and let <span class="math-inline"><math>c</math></span> be a cluster point of <span class="math-inline"><math>c</math></span>. Then <span class="math-inline"><math>\lim_{x \to c} f(x) = L</math></span> if and only if for all sequences <span class="math-inline"><math>(a_n)</math></span> from the domain <span class="math-inline"><math>A</math></span> where <span class="math-inline"><math>a_n \neq c</math></span> <span class="math-inline"><math>\forall n \in \mathbb{N}</math></span> and <span class="math-inline"><math>\lim_{n \to \infty} a_n = c</math></span> then <span class="math-inline"><math>\lim_{n \to \infty} f(a_n) = L</math></span>.</td>
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<blockquote style="background: white; border: 1px solid black; padding: 1em;">
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<td><strong>Theorem 1 (The Sequential Criterion for a Limit of a Function):</strong> Let <math>f : A \to \mathbb{R}</math> be a function and let <math>c</math> be a cluster point of <math>A</math>. Then <math>\lim_{x \to c} f(x) = L</math> if and only if for all sequences <math>(a_n)</math> from the domain <math>A</math> where <math>a_n \neq c</math> <math>\forall n \in \mathbb{N}</math> and <math>\lim_{n \to \infty} a_n = c</math> then <math>\lim_{n \to \infty} f(a_n) = L</math>.</td>
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<p>Consider a function <span class="math-inline">$f$</span> that has a limit <span class="math-inline">$L$</span> when <span class="math-inline">$x$</span> is close to <span class="math-inline">$c$</span>. Now consider all sequences <span class="math-inline">$(a_n)$</span> from the domain <span class="math-inline">$A$</span> where these sequences converge to <span class="math-inline">$c$</span>, that is <span class="math-inline">$\lim_{n \to \infty} a_n = c$</span>. The Sequential Criterion for a Limit of a Function says that then that as <span class="math-inline">$n$</span> goes to infinity, the function <span class="math-inline">$f$</span> evaluated at these <span class="math-inline">$a_n$</span> will have its limit go to <span class="math-inline">$L$</span>.</p>
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<p>Consider a function <math>f</math> that has a limit <math>L</math> when <math>x</math> is close to <math>c</math>. Now consider all sequences <math>(a_n)</math> from the domain <math>A</math> where these sequences converge to <math>c</math>, that is <math>\lim_{n \to \infty} a_n = c</math>. The Sequential Criterion for a Limit of a Function says that then that as <math>n</math> goes to infinity, the function <math>f</math> evaluated at these <math>a_n</math> will have its limit go to <math>L</math>.</p>
<p>For example, consider the function <span class="math-inline">$f: \mathbb{R} \to \mathbb{R}$</span> defined by the equation <span class="math-inline">$f(x) = x$</span>, and suppose we wanted to compute <span class="math-inline">$\lim_{x \to 0} x$</span>. We should already know that this limit is zero, that is <span class="math-inline">$\lim_{x \to 0} x = 0$</span>. Now consider the sequence <span class="math-inline">$(a_n) = \left ( \frac{1}{n} \right)$</span>. This sequence <span class="math-inline">$(a_n)$</span> is clearly contained in the domain of <span class="math-inline">$f$</span>. Furthermore, this sequence converges to 0, that is <span class="math-inline">$\lim_{n \to \infty} \frac{1}{n} = 0$</span>. If all such sequences <span class="math-inline">$(a_n)$</span> that converge to <span class="math-inline">$0$</span> have the property that <span class="math-inline">$(f(a_n))$</span> converges to <span class="math-inline">$f(0) = 0$</span>, then we can say that <span class="math-inline">$\lim_{n \to 0} f(x) = 0$</span>.</p>
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<p>For example, consider the function <math>f: \mathbb{R} \to \mathbb{R}</math> defined by the equation <math>f(x) = x</math>, and suppose we wanted to compute <math>\lim_{x \to 0} x</math>. We should already know that this limit is zero, that is <math>\lim_{x \to 0} x = 0</math>. Now consider the sequence <math>(a_n) = \left ( \frac{1}{n} \right)</math>. This sequence <math>(a_n)</math> is clearly contained in the domain of <math>f</math>. Furthermore, this sequence converges to 0, that is <math>\lim_{n \to \infty} \frac{1}{n} = 0</math>. If all such sequences <math>(a_n)</math> that converge to <math>0</math> have the property that <math>(f(a_n))</math> converges to <math>f(0) = 0</math>, then we can say that <math>\lim_{n \to 0} f(x) = 0</math>.</p>
 
<p>We will now look at the proof of The Sequential Criterion for a Limit of a Function.</p>
 
<p>We will now look at the proof of The Sequential Criterion for a Limit of a Function.</p>
 
<ul>
 
<ul>
<li><strong>Proof:</strong> <span class="math-inline">$\Rightarrow$</span> Suppose that <span class="math-inline">$\lim_{x \to c} f(x) = L$</span>, and let <span class="math-inline">$(a_n)$</span> be a sequence in <span class="math-inline">$A$</span> such that <span class="math-inline">$a_n \neq c$</span> <span class="math-inline">$\forall n \in \mathbb{N}$</span> such that <span class="math-inline">$\lim_{n \to \infty} a_n = c$</span>. We thus want to show that <span class="math-inline">$\lim_{n \to \infty} f(a_n) = L$</span>.</li>
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<li><strong>Proof:</strong> <math>\Rightarrow</math> Suppose that <math>\lim_{x \to c} f(x) = L</math>, and let <math>(a_n)</math> be a sequence in <math>A</math> such that <math>a_n \neq c</math> <math>\forall n \in \mathbb{N}</math> such that <math>\lim_{n \to \infty} a_n = c</math>. We thus want to show that <math>\lim_{n \to \infty} f(a_n) = L</math>.</li>
 
</ul>
 
</ul>
 
<ul>
 
<ul>
<li>Let <span class="math-inline">$\epsilon &gt; 0$</span>. We are given that <span class="math-inline">$\lim_{x \to c} f(x) = L$</span> and so for <span class="math-inline">$\epsilon &gt; 0$</span> there exists a <span class="math-inline">$\delta &gt; 0$</span> such that if <span class="math-inline">$x \in A$</span> and <span class="math-inline">$0 &lt; \mid x - c \mid &lt; \delta$</span> then we have that <span class="math-inline">$\mid f(x) - L \mid &lt; \epsilon$</span>. Now since <span class="math-inline">$\delta &gt; 0$</span>, since we have that <span class="math-inline">$\lim_{n \to \infty} a_n = c$</span> then there exists an <span class="math-inline">$N \in \mathbb{N}$</span> such that if <span class="math-inline">$n N$</span> then <span class="math-inline">$\mid a_n - c \mid &lt; \delta$</span>. Therefore <span class="math-inline">$a_n \in V_{\delta} (c) \cap A$</span>.</li>
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<li>Let <math>\varepsilon > 0</math>. We are given that <math>\lim_{x \to c} f(x) = L</math> and so for <math>\varepsilon > 0</math> there exists a <math>\delta > 0</math> such that if <math>x \in A</math> and <math>0 < \mid x - c \mid < \delta</math> then we have that <math>\mid f(x) - L \mid < \varepsilon</math>. Now since <math>\delta > 0</math>, since we have that <math>\lim_{n \to \infty} a_n = c</math> then there exists an <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> then <math>\mid a_n - c \mid < \delta</math>. Therefore <math>a_n \in V_{\delta} (c) \cap A</math>.</li>
 
</ul>
 
</ul>
 
<ul>
 
<ul>
<li>Therefore it must be that <span class="math-inline">$\mid f(a_n) - L \mid &lt; \epsilon$</span>, in other words, <span class="math-inline">$\forall n N$</span> we have that <span class="math-inline">$\mid f(a_n) - L \mid &lt; \epsilon$</span> and so <span class="math-inline">$\lim_{n \to \infty} f(a_n) = L$</span>.</li>
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<li>Therefore it must be that <math>\mid f(a_n) - L \mid < \varepsilon</math>, in other words, <math>\forall n \geq N</math> we have that <math>\mid f(a_n) - L \mid < \varepsilon</math> and so <math>\lim_{n \to \infty} f(a_n) = L</math>.</li>
 
</ul>
 
</ul>
 
<ul>
 
<ul>
<li><span class="math-inline">$\Leftarrow$</span> Suppose that for all <span class="math-inline">$(a_n)$</span> in <span class="math-inline">$A$</span> such that <span class="math-inline">$a_n \neq c$</span> <span class="math-inline">$\forall n \in \mathbb{N}$</span> and <span class="math-inline">$\lim_{n \to \infty} a_n = c$</span>, we have that <span class="math-inline">$\lim_{n \to \infty} f(a_n) = L$</span>. We want to show that <span class="math-inline">$\lim_{x \to c} f(x) = L$</span>.</li>
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<li><math>\Leftarrow</math> Suppose that for all <math>(a_n)</math> in <math>A</math> such that <math>a_n \neq c</math> <math>\forall n \in \mathbb{N}</math> and <math>\lim_{n \to \infty} a_n = c</math>, we have that <math>\lim_{n \to \infty} f(a_n) = L</math>. We want to show that <math>\lim_{x \to c} f(x) = L</math>.</li>
 
</ul>
 
</ul>
 
<ul>
 
<ul>
<li>Suppose not, in other words, suppose that <span class="math-inline">$\exists \epsilon_0 &gt; 0$</span> such that <span class="math-inline">$\forall \delta &gt; 0$</span> then <span class="math-inline">$\exists x_{\delta} \in A \cap V_{\delta} (c) \setminus \{ c \}$</span> such that <span class="math-inline">$\mid f(x_{\delta}) - L \mid \epsilon_0$</span>. Let <span class="math-inline">$\delta_n = \frac{1}{n}$</span>. Then there exists <span class="math-inline">$x_{\delta_n} = a_n \in A \cap V_{\delta_n} (c) \setminus \{ c \}$</span>, in other words, <span class="math-inline">$0 &lt; \mid a_n - c \mid &lt; \frac{1}{n}$</span> and <span class="math-inline">$\lim_{n \to \infty} a_n = c$</span>. However, <span class="math-inline">$\mid f(a_n) - L \mid \epsilon_0$</span> so <span class="math-inline">$\lim_{n \to \infty} f(a_n) \neq L$</span>, a contradiction. Therefore <span class="math-inline">$\lim_{x \to c} f(x) = L$</span>. <span class="math-inline">$\blacksquare$</span></li>
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<li>Suppose not, in other words, suppose that <math>\exists \varepsilon_0 > 0</math> such that <math>\forall \delta > 0</math> then <math>\exists x_{\delta} \in A \cap V_{\delta} (c) \setminus \{ c \}</math> such that <math>\mid f(x_{\delta}) - L \mid \geq \varepsilon_0</math>. Let <math>\delta_n = \frac{1}{n}</math>. Then there exists <math>x_{\delta_n} = a_n \in A \cap V_{\delta_n} (c) \setminus \{ c \}</math>, in other words, <math>0 < \mid a_n - c \mid < \frac{1}{n}</math> and <math>\lim_{n \to \infty} a_n = c</math>. However, <math>\mid f(a_n) - L \mid \geq \varepsilon_0</math> so <math>\lim_{n \to \infty} f(a_n) \neq L</math>, a contradiction. Therefore <math>\lim_{x \to c} f(x) = L</math>.  
 
</ul>
 
</ul>
  
==Resources==
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== Licensing ==  
* [http://mathonline.wikidot.com/the-sequential-criterion-for-a-limit-of-a-function The Sequential Criterion for a Limit of a Function]
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Content obtained and/or adapted from:
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* [http://mathonline.wikidot.com/the-sequential-criterion-for-a-limit-of-a-function The Sequential Criterion for a Limit of a Function, mathonline.wikidot.com] under a CC BY-SA license

Latest revision as of 14:11, 7 November 2021

We will now look at a very important theorem known as The Sequential Criterion for a Limit which merges the concept of the limit of a function at a cluster point from with regards to sequences from that converge to .

Theorem 1 (The Sequential Criterion for a Limit of a Function): Let be a function and let be a cluster point of . Then if and only if for all sequences from the domain where and then .

Consider a function that has a limit when is close to . Now consider all sequences from the domain where these sequences converge to , that is . The Sequential Criterion for a Limit of a Function says that then that as goes to infinity, the function evaluated at these will have its limit go to .

For example, consider the function defined by the equation , and suppose we wanted to compute . We should already know that this limit is zero, that is . Now consider the sequence . This sequence is clearly contained in the domain of . Furthermore, this sequence converges to 0, that is . If all such sequences that converge to have the property that converges to , then we can say that .

We will now look at the proof of The Sequential Criterion for a Limit of a Function.

  • Proof: Suppose that , and let be a sequence in such that such that . We thus want to show that .
  • Let . We are given that and so for there exists a such that if and then we have that . Now since , since we have that then there exists an such that if then . Therefore .
  • Therefore it must be that , in other words, we have that and so .
  • Suppose that for all in such that and , we have that . We want to show that .
  • Suppose not, in other words, suppose that such that then such that . Let . Then there exists , in other words, and . However, so , a contradiction. Therefore .

Licensing

Content obtained and/or adapted from: