# $f$ converges then $f'$ converges to $0$?

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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May the 4 downvoters who downvoted this question step forward. – Rudy the Reindeer Jan 19 '13 at 17:58

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 $x \to \infty$.