If $\lim_{x \rightarrow \infty} f(x)$ is finite, is it true that $ \lim_{x \rightarrow \infty} f'(x) = 0$? Does finite $\lim_{x \rightarrow \infty} f(x)$ imply that $\lim_{x \rightarrow \infty} f'(x) = 0$? If not, could you provide a counterexample?
It's obvious for constant function. But what about others?

Dan Shved counterexample provide a complete answer, however, if the limit of $\lim_{x\to\infty}f'(x)$ does exist, then $\lim_{x\to\infty}f'(x)$ must be equal to zero. To see this we can apply Lagrange's mean value theorem, we get 
$$\frac{f(x_n+h)-f(x_n)}{h}=f'(c_n)$$ 
where $c_n\in(x_n , x_n+h)$ for any monotone sequence $x_n$, where $\lim_{x\to\infty}x_n=\infty$ and  $h=1$. Since $\lim_{n\to\infty}c_n=\infty$ we get
$$\begin{array}{ll}
\lim_{n\to\infty}f'(c_n) &= \lim_{n\to\infty}\left(f(x_n+1)-f(x_n)\right) \\\\
                         &=0.
\end{array}$$
and this implies that if the limits of $\lim_{x\to\infty}f'(x)$ exists and
$\lim_{x\to\infty}f(x)$ is finite, then $\lim_{x\to\infty}f'(x)=0$. 
