3
$\begingroup$

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

  • 1
    If $\lim_{x \to \infty}f'(x)$ exists, there is no such function.2012-12-23

2 Answers 2

6

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

1

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}} which contradicts the mean value (Lagrange's) theorem. Therefore if $\lim\limits_{x\to+\infty}{\dfrac{df(x)}{dx}}$ exists, it equals $0.$