If $\lim_{x\to\infty} |f^\prime(x)|=0$ and $f^\prime(x)$ is continuous on $\Bbb R$, does this imply that $\lim_{x\to\infty} f(x)$ exists? [duplicate]
Solution 1:
First idea: If you drop the boundedness assumption, $\ln(x)$ does provide a counterexample to what you want.
Now think about how you can transform this counterexample into a bounded counterexample.
Hint
Typical bounded functions include $\sin$ and $\cos$.
Solution 2:
In general the answer is NOT. Let's consider the function $f(t) = \sin(\ln(t))$. Its derivative $f' = \cos(\ln(t))/t$ converges to $0$ but in limit, the function keeps oscillating.