If the limits at infinity of $f$ and $f'$ exist and are finite, then is $\lim_{x \rightarrow \infty} f'(x) = 0$?
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