Prove that $\lim_{x \rightarrow \infty} \frac{f(x)}{x}= \alpha$ when $f$ is continuous and $\lim_{x \rightarrow \infty} (f(x+1) - f(x)) = \alpha$
If $\lim_{x\to+\infty}[f(x+1)-f(x)]= \ell,$ then $\lim\limits_{x\to+\infty}\frac{f(x)}x=\ell$.