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Do there exist functions $f$, where $\lim_{x \rightarrow \infty} f(x) = 0$ and $\lim_{x \rightarrow \infty} \frac{df(x)}{dx} \neq 0$?

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If $\lim_{x \to \infty}f'(x)$ exists, there is no such function. – Mercy King Dec 23 '12 at 20:08
up vote 6 down vote accepted

Take $f(x)= \frac{1}{x} \sin(x^2)$.

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If exists $\lim\limits_{x\to+\infty}{\dfrac{df(x)}{dx}}=a\ne{0}$ then $\forall \varepsilon>0\;\;\exists x_\varepsilon\colon \;\; \forall {x>x_\varepsilon}$ $$a-\varepsilon<{\dfrac{df(x)}{dx}}<a+\varepsilon,$$ which contradicts the mean value (Lagrange's) theorem. Therefore if $\lim\limits_{x\to+\infty}{\dfrac{df(x)}{dx}}$ exists, it equals $0.$

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