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I have a homework question, looks simple but I can't figure out a way to solve it. Any clue or help will be helpful.

Let $f: [0,\infty)\rightarrow\mathbb R$ be a differentiable function such that $\lim_{x\rightarrow\infty}f'(x)=0$.

Prove or disprove: $\lim_{x\rightarrow\infty}f(x)$ exists (infinite limit is also considered as a limit).

Thanks a lot!

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  • $\begingroup$ Differentiable implies continuous. $\endgroup$
    – Wintermute
    Mar 28 '13 at 17:51
  • $\begingroup$ Look for a counterexample. The one I have in mind is messy, involves splicing. $\endgroup$ Mar 28 '13 at 17:54
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Picture a sinusoidal curve, but stretched in the horizontal direction more and more as $x\rightarrow\infty$. The amplitude is fixed but the velocity decreases. Something like $\sin (\sqrt{x})$.

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    $\begingroup$ Elegant and Brilliant! $\endgroup$
    – Simple.guy
    Apr 4 '13 at 16:38

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