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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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The statement is false.

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

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

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  • $\begingroup$ excellent!! thanks $\endgroup$
    – Analysis
    Aug 17 '14 at 21:22

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