From a exercise list:
Let $f:[a,+\infty)\rightarrow \mathbb{R}$ with continuous second derivative and such that $\lim\limits_{x\rightarrow\infty}f(x)=0$ and $\lim\limits_{x\rightarrow\infty}f''(x)=0$. Prove that $\lim\limits_{x\rightarrow\infty}f'(x)=0$.
Does anybody has a hint or a solution? I believe that, by the limit of $f$, $f'$ is either $0$ or unlimited, than the limit of $f''$ proves that the first option is the right one. But I don't know how to do it.
