Use mean value theorem to prove $ \lim_{x \to \infty}\left( f(x+1)-f(x) \right)=a ~$ as $ ~\lim_{ x \to \infty}f'(x)=a $
Suppose that $f$ is differentiable on $\mathbb{R}$ and $\lim_{x\to \infty}f'(x)=M$. Show that $\lim_{x\to \infty}f(x+1)-f(x)$ exists and find it.