# Math Help - LaGrange maybe...

1. ## LaGrange maybe...

Prove that if $f(x)$ differentiable on $(a, \infty)$ and if $lim_{x\to \infty} f'(x)=0$ then: $lim_{x\to \infty} \frac{f(x)}{x}=0$

2. Originally Posted by Also sprach Zarathustra
Prove that if $f(x)$ differentiable on $(a, \infty)$ and if $lim_{x\to \infty} f'(x)=0$ then: $lim_{x\to \infty} \frac{f(x)}{x}=0$
Assume $\lim_{x\to\infty}f(x)=\pm\infty$ otherwise this problem is trivial.

So as $x\to\infty$, $\frac{f(x)}{x}$ will have the indeterminate form $\frac{\infty}{\infty}$ so we can use L'hopital's rule: $\lim_{x\to\infty}\frac{f(x)}{x} = \lim_{x\to\infty}f'(x)=0$.

3. Super solution from a super Member!