Why doesn't $\lim_\limits{x\to\infty} f(x) = 0$ directly imply $\lim_\limits{x\to\infty}f'(x) = 0$

Question

I have a problem (which I had already asked as a question) but I still do not get the solution. The problem is that

Suppose $f$ is twice differentiable function on $(0,\infty)$ and $f''$ is bounded on $(0,\infty)$ and $\lim_\limits{x\to\infty}f(x) = 0$. Then prove that

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

Here I do not understand why $f''$ is required to be bounded i.e. why $f'$ is required to be uniformly continuous? As the limit of $f$ as $ x\to \infty$ is constant, the implication is that $f'$ must tend to $0$. I know my argument is wrong but I did not get counterexample. Any help will be appreciated.

Answer 1

As counterexample let consider

$$f(x)=\frac{\sin x^2}{x}\implies f'(x)=2\cos x^2-\frac{\sin x^2}{x^2}$$