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

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The Sequential Criterion for a Limit of a Function
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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>
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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>
 
<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 <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>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 <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>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>
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== Licensing ==  
 
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Content obtained and/or adapted from:
==Resources==
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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
* [http://mathonline.wikidot.com/the-sequential-criterion-for-a-limit-of-a-function The Sequential Criterion for a Limit of a Function]
 

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: