I need to prove or disprove this:
If $f$ is differentiable on $(1,\infty)$ and \lim\limits_{x\to\infty}f'(x)=L\lt \infty, then $\lim\limits_{x\to\infty}f(x)=\ell\leq\infty$.
After I didn't find any function to disprove with,
I started to think that if \lim\limits_{x\to\infty }f'(x)=L<\infty so f' is bounded and therfore $f$ is uniformly continuous but it doesn't mean that $\lim\limits_{x\to\infty }f(x)=l\leq\infty$, for example : $\sin x$ which is uniformly continuous, but it's limit when $x\to\infty$ does not exist.
What do you think?
Thank you.