limits - Does any $f(x)$ which for large $x$ grows slower than any power of $x$ necessarily satisfy $lim_{xtoinfty}f'(x)=0$?

The title is essentially the question. Intuitively I would await that any function $f(x)$, which for large $x$ grows slower than any power of $x$, must necessarily satisfy

$\lim_{x\to\infty}f'(x)=0$.

This holds at least for prominent examples like $\ln(x)$ and $\sqrt{x}$. But I don't know of a statement that it holds in general. Does anyone know or have a proof or counter examples?

Thanks for help!

EDIT: I forgot to mention that $f$ should also be strictly monotonically increasing, for large $x$, and also that it should be smooth ($C^\infty(\mathbb R)$).

Answer

I assume that $f$ is differentiable (otherwise $f'(x)$ is meaningless).

Assuming the limit exists, it must be $0$.

Otherwise, if

$$\lim_{x \to +\infty} f'(x)=L >0$$
then there exists $x_0 >0$ such that for $x>x_0$ necessarily
$$f'(x) > \frac{L}{2}$$ This implies that $$f(x)=f(x_0)+ \int_{x_0}^x f'(t) \mathrm dt>f(x_0)+\frac{L}{2}(x-x_0)$$
which contradicts our hypothesis that $f$ grows slower than $x$.

However, nothing ensures that the limit exists. As a counterexample, you could consider

$$f(x) = \int_0^x |\sin t|^t \mathrm dt$$