Question about $f$ continuous function with these conditions?

Question

Suppose I have a differentiable and bounded function $$f: [0, + \infty) \longrightarrow \mathbb{R}$$ such that $$\forall x \in [0, + \infty) \, : f(x) \cdot f'(x) > \sin x.$$

The question is: does it follow that $$\lim_{x\rightarrow \infty} f(x)$$necessarily exists?

Thanks for any suggestions and comments, even modify some conditions of the problem.

Answer 1

The function $g(u) =(f(u))^2 +2\cdot \cos u $ is increasing and bounded hence there exists $\lim_{u\to \infty } g(u) $ hence $\lim_{u\to \infty } f(u) $ cannot exist.