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Let $f: \mathbb{R} \to \mathbb{R}$ such that:

(i) $f$ is differentiable in $\mathbb{R}$

(ii) The limit $\lim_{x\to \infty}f(x)$ doesn't exist (EDIT : neither finite nor infinite).

Show that there is $x_0\in\mathbb{R}$ such that $f'(x_0)=0$.

I understand intuitively that these terms satisfy that the function is not injective and thus I can use lagrange, but I'm wondering how to show it formally.

Any help would be appreciated

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If $f'$ has no zeros, it follows from Darboux's theorem that either $f'(x)$ is always greater than $0$ or always smaller. Therefore, $f$ is either increasing or decreasing. But then $\lim_{x\to\infty}f(x)$ must exist (in $\mathbb{R}\cup\{+\infty,-\infty\}$).

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  • $\begingroup$ Exactly what I was looking for. Thank you! $\endgroup$ – Moshe King Jan 25 '18 at 17:12
  • $\begingroup$ Just a small question. If the function is increasing, why does that imply there's a limit in infinity? @José Carlos Santos $\endgroup$ – Moshe King Jan 25 '18 at 17:15
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    $\begingroup$ @GADI Because either the function is unbounded above, in which case $\lim_{x\to+\infty}f(x)=+\infty$, or it is, in which case $\lim_{x\to+\infty}f(x)=\sup\{f(x)\,|\,x\in\mathbb{R}\}$. $\endgroup$ – José Carlos Santos Jan 25 '18 at 17:17
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$$f(x) = x$$

  • Differentiable function

  • the limit $\lim_{x \to \infty} f(x) $ doesn't exist.

There is no such $x_ 0 ∈ \Bbb R$ such that $f′(x_0)=0$.

Contradiction.

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    $\begingroup$ I think by doesnt exist he means like cosine and sine functions $\endgroup$ – Atmos Jan 25 '18 at 17:06
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    $\begingroup$ Based on the context, I'm guessing (almost sure...) that $\lim_{x \to \infty} f(x) = \infty$ counts as the limit existing for the asker. $\endgroup$ – user296602 Jan 25 '18 at 17:07
  • $\begingroup$ @user296602 ...That's unfair :( $\endgroup$ – Jaideep Khare Jan 25 '18 at 17:08
  • $\begingroup$ The OP needs to clarify that. When the answer is infinity, the limit does not exist if you want to pursue limit laws. Otherwise, Jaideep has a valid point here $\endgroup$ – imranfat Jan 25 '18 at 17:09
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    $\begingroup$ To everybody: Some authors accept infinity as an appropriate answer, others not. The thing is, when discussing issues like "curve end- behaving", one can loosely use infinity as an answer so that the reader knows "Ah, there is no horizontal asymptote" or something like that. When embarking on more algebra, and limit laws come into play, infinity is no longer acceptable as an answer as limit laws no longer can be blindly applied, that is, caution needs to be applied. Hence, the "rules of the game" should be stated before hand. $\endgroup$ – imranfat Jan 25 '18 at 17:17

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