Is it true that f is differentiable, and $\lim_{x\to\infty}f(x)=0,$ then $\lim_{x\to\infty} f'(x)=0$ Is it true that 
If $f\in C^1(\mathbb{R})$ and $\lim_{x\to\infty}f(x)=0$, then $\lim_{x\to\infty} f'(x)=0$.

Nope. Take $f(x) = \frac{\sin(x^2)}{x}$.
(edit: and define $f(0) = 0$)
