How to prove $\lim\limits_{x\rightarrow\infty}f(x)=0$ and $\lim\limits_{x\rightarrow\infty}f''(x)=0\Rightarrow\lim\limits_{x\rightarrow\infty}f'(x)$?
If $f(x)\to 0$ as $x\to\infty$ and $f''$ is bounded, show that $f'(x)\to0$ as $x\to\infty$