# If $f$ and $f'$ have a limit then is it equal to zero for $f'$? [duplicate]

I have this question :

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ , and $f\in C^1(\mathbb{R})$ such that $$\begin{cases}\lim_{t\to\infty} f(t)=l_1\\ \lim_{t\to\infty}f'(t)=l_2\end{cases}$$ $l_1,\, l_2\in\mathbb R$

How to prove that $l_2=0$?

## marked as duplicate by Hans Lundmark, user91500, user228113, John B, WatsonMar 26 '16 at 10:22

By the mean value theorem, for each $n\in\Bbb N$ there exists $c_n\in[n,n+1]$ such that $$f(n+1)-f(n)=f'(c_n).$$ Now, take the limit.
• spenser ,,, do $c_n$ necessary go to $\infty$ when we take the limite ??$– M.luffy Mar 26 '16 at 9:12 • @M.luffy Yes because$c_n\geq n$for all$n\$. – Spenser Mar 26 '16 at 9:12