I've been trying to solve the following problem:
Suppose that $f$ and f' are continuous functions on $\mathbb{R}$, and that $\displaystyle\lim_{x\to\infty}f(x)$ and \displaystyle\lim_{x\to\infty}f'(x) exist. Show that \displaystyle\lim_{x\to\infty}f'(x) = 0.
I'm not entirely sure what to do. Since there's not a lot of information given, I guess there isn't very much one can do. I tried using the definition of the derivative and showing that it went to $0$ as $x$ went to $\infty$ but that didn't really work out. Now I'm thinking I should assume \displaystyle\lim_{x\to\infty}f'(x) = L \neq 0 and try to get a contradiction, but I'm not sure where the contradiction would come from.
Could somebody point me in the right direction (e.g. a certain theorem or property I have to use?) Thanks