Mean Value Theorem

Question

Prove that if $f$ is a differentiable function on $(0,\infty)$ and $f$ and $f'$ both have finite limits at infinity, then lim as $x$ goes to infinity of $f'(x) = 0$. Hint: apply the Mean Value Theorem to $f$ for large values of $a$ and $b$.

Answer 1

Let $L=\lim_{x\rightarrow+\infty}f(x)$, there exists $N$ such that $x>N$ implies that $\mid f(x)-L\mid <1$, for every integer $n>N$, $f(n+n)-f(n)=f'(c_n)n$, $c_n\in [2n,n]$.

This implies that $\mid f(n+n)-f(n)\mid =n\mid f'(c_n)\mid <1$ this implies that $\mid f'(c_n)\mid< 1/n$, since $\lim_nc_n=+\infty$ and $\lim_{x\rightarrow +\infty}f'(x)$ exists, you deduce that $\lim_{x\rightarrow +\infty}f'(x)=0$.