Real Function Limits:Sequential Criterion

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The Sequential Criterion for a Limit of a Function

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 .


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