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Let $f: \mathbb R \rightarrow \mathbb R$ continuous and differentiable. Assume that $$ \lim_{x \rightarrow \infty} f(x) = y_0 \in \mathbb R $$

Is it then true that $$ \lim_{x \rightarrow \infty} f'(x) = 0 $$ ?

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    $\begingroup$ May the 4 downvoters who downvoted this question step forward. $\endgroup$ – Rudy the Reindeer Jan 19 '13 at 17:58
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Take $f(x) = \frac{1}{x}\sin(x^2)$ which converges to zero as $x \to \infty$ but $f^{'}(x) = \frac{-1}{x^2} \sin(x^2) + 2\cos(x^2)$ does not converge as $x \to \infty$.

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  • $\begingroup$ Yeah thanks. It was just and intuitive idea, so thanks for this example. $\endgroup$ – user42761 Jan 14 '13 at 22:41
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    $\begingroup$ Why does the question have so many down votes?? $\endgroup$ – Adam Rubinson Jan 15 '13 at 14:04
  • $\begingroup$ I was confused, too. Next time I will think more deeply about my questions, first. $\endgroup$ – user42761 Jan 19 '13 at 18:05

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