the limit of the differential of a function is 0 if the limit of this function is a constant

Question

Let $f$ be a positive function defined on $(0,+\infty)$ and $\lim_{x\rightarrow \infty }f(x)=c_{0},0\leq c_{0}<\infty $. $f$ is absolutely continuous.

Does $\lim_{x\rightarrow \infty }f'(x)=0$ hold?

If so, is there any easy proof rather than using the definition of limit?

Answer 1

The answer is in fact no. Consider the function $f(x) = \frac{1}{x+1} \sin(e^x)+2$. This is absolutely continuous, but the derivative does not converge as exponentials grow faster than any power.

Answer 2

I don't think that it holds.

Take $f(x)=\frac{\sin(x^2)}{x}$.

$\lim_{x\to\infty}f(x)=0$ but $f'(x)=\log(x)\sin(x^2)+2\cos(x^2)$ doesn't converge.