If the limit of a derivative is zero as $x \to \infty$, what can we say about $f(x+1)-f(x)$?
If $f:\mathbb{R}\rightarrow \mathbb{R}$ is differentiable and $\lim_{x\to \infty } f^\prime(x)=0$ show that $\lim _{x\to \infty } (f(x+1)-f(x))=0$.