Proving the limit at $\infty$ of the derivative $f'$ is $0$ if it and the limit of the function $f$ exist.
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