TITLE: How to prove convergent function imply its derivative equals to zero?
QUESTION [5 upvotes]: Let $f\colon (0,\infty) \to\Bbb R$ be differentiable and let $A$ and $B$ be real numbers. 
Prove that if $f(t) \to A$ and $f′(t) \to B$ as $t \to \infty$ then $B = 0$.

REPLY [7 votes]: By the Mean Value Theorem, for every positive integer $n$, 
$$\frac{f(n+1)-f(n)}{1}=f'(c_n),$$
for some $c_n$ between $n$ and $n+1$. Since $f(n+1)-f(n)\to 0$, $f'(c_n)$ must have small absolute value when $n$ is large. So if the limit of $f'(x)$ exists, it must be $0$.