Prove limit of function

Question

Let $f:(a,+\infty) \to \mathbb{R}$ and on every finite $(a,b)$ interval function $f$ is bounded. Then $$\lim_{x \to \infty}\frac{f(x)}{x}=\lim_{x \to \infty}f(x+1)-f(x)$$

How can we prove or disprove this statement?

Answer 1

The statement is false.

Consider $f(x)=\sin \frac{\pi}{2} x$.

The LHS goes to $0$, while the RHS does not converge.