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?
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Sign up to join this communityThe statement is false.
Consider $f(x)=\sin \frac{\pi}{2} x$.
The LHS goes to $0$, while the RHS does not converge.