The Limit Theorems for Functions

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The Uniqueness of Limits of a Function Theorem

Recall from The Limit of a Function page that for a function where is a cluster point of , then if such that if and then . We have not yet established that the limit is unique, so is it possible that and where ? The following theorem will show that this cannot happen.

Theorem (Uniqueness of Limits): Let be a function and let be a cluster point of . Then if are both limits of at , that is and , then .
  • Proof: Let be a function and let be a cluster point of . Also let and . Suppose that . We will show that this leads to a contradiction. Let be given.
  • Since , then for such that if and then .
  • Similarly, since then for such that if and then . Now let and so we have that:

  • But is arbitrary, so this implies that , that is , a contradiction. So our assumption that was false, and so if then is unique.