If $f(x)$ is a real function with (1) $\lim\limits_{x\to+\infty} f(x)=c\in\mathbb{R}$ and if I suppose that $\lim\limits_{x\to+\infty} f'(x)$ exist, then is it true that $\lim\limits_{x\to+\infty} f'(x)=0$? Furthermore, if I suppose $f\in C^1(\mathbb{R})$ and hypothesis (1), then $\lim\limits_{x\to+\infty} f'(x)=0$? <!——>
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It is not clear what the question is in the "furthermore" part. FWIW being $C^1$ and having a horizontal asymptote at $+\infty$ is not enough to guarantee that $f'$ has a limit at $+\infty$. – dxiv Jul 06 '21 at 17:23
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https://math.stackexchange.com/questions/42277/proving-that-lim-limits-x-to-inftyfx-0-when-lim-limits-x-to-inftyf, https://math.stackexchange.com/questions/162078/if-a-function-has-a-finite-limit-at-infinity-does-that-imply-its-derivative-goe – Hans Lundmark Jul 06 '21 at 17:41
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“then is true that $\lim\limits_{x\to+\infty} f'(x)=0$?”—I think that would make sense. Your first statement says the function converges to $c$, and the next supposes the limit for the derivative exists. Since the function converges to single value, it’s slope must level off, meaning it’s derivative converges to $0$. Sorry I haven’t said this mathematically, I’m not sure how to do that. – Cotton Headed Ninnymuggins Jul 06 '21 at 17:45
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HINT:
From the mean-value theorem, we have
$$f(x+1)-f(x)=f'(\xi)$$
where $\xi\in(x,x+1)$. Can you finish now?
Mark Viola
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@user910194 Please let me know how I can improve my answer. I really want to give you the best answer I can. – Mark Viola Sep 25 '21 at 16:56
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Here's a different approach. Let $a=\lim_{x\to\infty}f'(x)$. Then by l'Hôpitâl (for instance), we have $$\lim_{x\to\infty}\frac{f(x)}{x}=a.$$ If $a\neq 0$, $f$ wouldn't have a horizontal asymptote.
Zuy
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