is it true that a continuously differentiable function $f: [0,+\infty) \rightarrow \mathbb{R}^n$ that is bounded and $\|{f'(x)}\|\leq \frac{C}{x} \ \forall x \in [0,\infty)$ has limit for $x \rightarrow +\infty$ ?
Because it seems to me that it is true but I can't prove it.
$f(x)=\sin (\ln (1+x))$ is a counter-example. Note that $\lim_{n\to \infty}f(e^{\frac {(2n+1)\pi} 2}-1)$ does not exist. The hypothesis holds with $C=1$.