Prove that if $\lim\limits_{x\to\infty}f(x)$ and \lim\limits_{x\to\infty}f''(x) exist, then \lim\limits_{x\to\infty}f'(x)=0.
I can prove that \lim\limits_{x\to\infty}f''(x)=0. Otherwise f'(x) goes to infinity and $f(x)$ goes to infinity, contradicting the fact that $\lim\limits_{x\to\infty}f(x)$ exists. I can also prove that if \lim\limits_{x\to\infty}f'(x) exists, it must be 0. So it remains to prove that \lim\limits_{x\to\infty}f'(x) exists. I'm stuck at this point.