analysis question for limits of functions

mlm823 · **Joined:** Sat, 15 Nov 2008 06:23:03 UTC **Posts:** 18

I have this statement:

If f does not have a limit at c, then there exists a sequence s(n) in D with each s(n)≠c such that s(n) converges to c, but (f(s(n)) is divergent.

I know that it is true but not quite sure how to justify my answer. It is obvious that if f doesn't have a limit at c then it diverges -- but how to show that the sequence of that function diverges?

Shadow · **Posted:** Wed, 19 Nov 2008 22:28:44 UTC

First of all what are all these things you're asking about? What are

and

? There may not be such a sequence in general, say $D=(0,1)\cup \{2\}$ . Then

cannot possibly have a limit at c=2

since we cannot get arbitrarily close to it. But there's no sequence where $\exists N\in\mathbb{N}$ such that $\forall n>N,\;s(n)\ne c$ when $s(n)\rightarrow c$ .

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analysis question for limits of functions

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