# Proving that lim\lim\limits_{x\to\infty}f'(x) = 0 when \lim\limits_{x\to\infty}f(x)\lim\limits_{x\to\infty}f(x) and \lim\limits_{x\to\infty}f'(x)\lim\limits_{x\to\infty}f'(x) exist

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

Apply a L’Hospital slick trick: $\,$ if $\rm\ f + f\,'\!\to L\$ as $\rm\ x\to\infty\$ then $\rm\ f\to L,\ f\,'\!\to 0,\,$ since