Does $\lim_{x\to\infty} f'(x) = 0$ implies $\lim_{x\to\infty} f(x)$ exists finite?

Question

If $f(x)$ is differentiable for $x>0$ and $$\lim_{x\to\infty}f'(x)=0,$$ then $$\lim_{x\to\infty}f(x)$$ exists finite.

Is that statement right or wrong?

Answer 1

First of all, your limits are strange. You might like to write $$ \lim_{x\to\infty} f'(x)=0\text{ and }\lim_{x\to\infty} f(x)=L. $$

Next, we are not doing your homework. You should try it yourself.

The first try should be $f'(x)=x^{-n}$ for $n\in\mathbb N$. Obviously, $\lim_{x\to\infty} f'(x)=0$. How about $f$?

(Be carefull to check $n=1$ and $n>2$ as different cases)