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Clarification: Does $\lim\limits_{x \to \infty}f'(x)=0$ as $x$ approaches infinity mean $\lim\limits_{x \to \infty} f(x)$ exists in the extended real numbers $[-\infty,\infty]$?

I was thinking a lot about this problem and I couldn't prove it; I would appreciate any help!

This isn't a duplicate! In that problem you have to prove that the limit of f(x) exists in the extended real numbers, you don't assume it does.

Tamir
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No, there is a simple counter example. $$f'(x)=\frac1x\implies \lim_{x\to\infty}f'(x)=0$$ But $$f(x)=\log x, \lim_{x\to\infty}f(x)=\infty$$

John Doe
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    he asked about $\lim \limits_{x \to 0} f(x)$, furthermore he asks what is the general behavior. – Ahmad Oct 07 '17 at 20:57
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    @Ahmad Hmm, I am sure he must have meant $\infty$, especially after reading the title. If he did mean $x\to0$, then you could take $f(x)=\frac1x$ to get infinite behaviour at $0$, and something like $f(x)=e^{-x}$ to get finite behaviour. But as mentioned in the comment on the question, you can combine functions to get any behaviour at $0$, regardless of what happens at $\infty$ – John Doe Oct 07 '17 at 21:01
  • Please read again the problem, somewhy someone edited it wrongly... – Tamir Oct 07 '17 at 21:06
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    @Tamir I answered your question with $x\to\infty$. My comment referred to $x\to 0$. – John Doe Oct 07 '17 at 21:07
  • where did you answer it with x goes to infinity? – Tamir Oct 07 '17 at 21:12
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    @Tamir I gave you an example of a function $f(x)$ whose derivative goes to $0$ as $x\to\infty$, but where the actual function $f(x)$ does not converge to a finite value - it goes to infinity. – John Doe Oct 07 '17 at 21:14
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    right, but I want to prove that f(x) converges to a finite or infinite value! – Tamir Oct 07 '17 at 21:19
  • @Tamir My answer shows that $f(x)$ could be infinite as $x\to\infty$. There are many different examples where $f(x)$ could converge to any finite value. e.g. $f(x)=\frac1x+a$ converges to any finite value $a$ that you want it to. – John Doe Oct 07 '17 at 22:27
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    @John Doe Tamir wants to know if the limit exists in general (including possibly $\pm \infty$). See my edit of the question. – D_S Oct 07 '17 at 22:55