Does $f(x)$ converge when $x$ goes to infinity if $f'(x)$ goes to $0$ when $x$ goes to infinity?
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