$\underset{x\rightarrow\infty}{\lim}\frac{f(x)}{x}=0$ Implies $\underset{x\rightarrow\infty}{f'(x)}=0$
Proving that $\lim\limits_{x\to\infty}f'(x) = 0$ when $\lim\limits_{x\to\infty}f(x)$ and $\lim\limits_{x\to\infty}f'(x)$ exist