limits: prove or disprove

Question

I need to prove / disprove the following:

Let $ f(x) $ be a differentiable function in $ [a,\infty) $ for $ a \in R $ and $ \lim_{x \to \infty } f(x) = L $ then $ \lim_{x \to \infty } f'(x) = 0 $

all I gotis my intuition that it isn't true since there might be some sinus style function that for each $ \epsilon$ will have points more the differentiable isn't 0...

Thank you

Answer 1

This is a couter example :

The function $$ f(x) = \frac{1}{x} \sin(x^2) $$ is differentiable in $[1,+\infty)$ and satisfies $$ \lim_{x \rightarrow +\infty} f(x) = 0. $$

However, its derivative is given by $$ f'(x) = 2\cos(x^2) - \frac{\sin(x^2)}{x^2} $$ and its limit is not defined at $+\infty$.