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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$?

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

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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.

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  • $\begingroup$ spenser ,,, do $c_n$ necessary go to $\infty$ when we take the limite ??$ $\endgroup$ – M.luffy Mar 26 '16 at 9:12
  • $\begingroup$ @M.luffy Yes because $c_n\geq n$ for all $n$. $\endgroup$ – Spenser Mar 26 '16 at 9:12
  • $\begingroup$ My pleasure. :-) $\endgroup$ – Spenser Mar 26 '16 at 9:13

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