Can someone example and give an example? Given an example of a function $f $ such that $\lim_{x\rightarrow \infty } f(x)$   exists,  but   $\lim_{x\rightarrow \infty } f'(x)$  does not exist. 

Consider
$$
f(x) = \frac{\sin(x^2)}{x}.
$$
Using the Squeeze Theorem, you can show that
$$
\lim_{x \to \infty} f(x) = 0.
$$
However, its derivative
$$
f'(x) = 2\cos(x^2) - \frac{\sin(x^2)}{x^2}
$$
never settles down as $x \to \infty$.
