Prove or find a counterexample

Question

Suppose $f(x)$ is bounded and differentiable on $[0,\infty)$. Consider the statement:

If $\lim_{x\to \infty}f(x)=0 $, then $\lim_{x\to \infty}f'(x)=0 $.

Prove it if it's right or show a counterexample.

I think it's not true and trying to find a counterexample by using functions containing $\cos(\frac{1}{x})$ or $e^{-x}$ or other things. The final goal is to find something make it's derivative' limit at infite doesn't exist. Since if it exists, it must equals zero or $f(x)$ will not be bounded.

Answer 1

You can take $f(x)=e^{-x}\sin(e^x)$, for instance. Note that $f'(x)=-e^{-x}\sin(e^x)+\cos(e^x)$ and that therefore the limite $\lim_{x\to+\infty}f'(x)$ does not exist.