2
$\begingroup$

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!

$\endgroup$
  • $\begingroup$ 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. $\endgroup$ – André Nicolas Mar 28 '13 at 17:54
6
$\begingroup$

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 $\sin (\sqrt{x})$.

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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.