$f$ converges then $f'$ converges to $0$? Let $f: \mathbb R \rightarrow \mathbb R$ continuous and differentiable. Assume that 
$$
 \lim_{x  \rightarrow \infty} f(x) = y_0 \in \mathbb R
$$
Is it then true that 
$$
 \lim_{x \rightarrow \infty} f'(x) = 0 
$$ ?

Take $f(x) = \frac{1}{x}\sin(x^2)$ which converges to zero as $x \to \infty$ but $f^{'}(x) = \frac{-1}{x^2} \sin(x^2) + 2\cos(x^2)$ does not converge as $x \to \infty$.
