Is it true that f is differentiable, and $\lim_{x\to\infty}f(x)=0,$ then $\lim_{x\to\infty} f'(x)=0$

Question

Is it true that

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

Answer 1

Nope. Take $f(x) = \frac{\sin(x^2)}{x}$.

(edit: and define $f(0) = 0$)