Differentiable function and limit

Question

$f(x)$ is a differentiable function such that both $T=lim_{x\to\infty} f(x)$ and $L=lim_{x\to\infty} f'(x)$ exist and are finite. Then must it be true that $L=0$? Seems true for many functions. But I am having hard time actually formally proving this. Thanks for any help.

Answer 1

$f(x+1)-f(x) = \int_x^{x+1}f'(x)$. Since $lim_{+\infty}f(x) <+\infty$, for every $c>0$, there exists $n$ such that $x,y\geq n$ implies that $\mid f(y)-f(x)\mid <c$, in particular $\mid f(n+1)-f(n)\mid = \mid \int_n^{n+1}f'(x)\mid =\mid f'(c_n)\mid<c$ where $c_n\in [n,n+1]$ by the mean value theorem. This implies $lim f'(c_n) =lim_{+\infty} f'(x)= 0$.