# Lim f'(x)=0 implies lim f(x)/x = 0

## Homework Statement

Given that the function $f$ is differentiable on the interval $(a,\infty)$ and that $\lim_{x\rightarrow\infty}f'(x)= 0$. Show that $\lim_{x\rightarrow\infty}\frac{f(x)}{x}= 0$.

## The Attempt at a Solution

I have a pretty good intuition of why this is true: As f(x) approaches infinity, its derivative approaches 0. This means that, at some point, the function slope becomes less than that of x.
This means that when x approaches infinity, x is dominant over f(x) and thus f(x)/x => 0.

I can't come up with any strict mathematical proof though, so I would be very greatful if someone could help me with this.

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Mark44
Mentor

## Homework Statement

Given that the function $f$ is differentiable on the interval $(a,\infty)$ and that $\lim_{x\rightarrow\infty}f'(x)= 0$. Show that $\lim_{x\rightarrow\infty}\frac{f(x)}{x}= 0$.

## The Attempt at a Solution

I have a pretty good intuition of why this is true: As f(x) approaches infinity, its derivative approaches 0. This means that, at some point, the function slope becomes less than that of x.
This means that when x approaches infinity, x is dominant over f(x) and thus f(x)/x => 0.

I can't come up with any strict mathematical proof though, so I would be very greatful if someone could help me with this.
L' Hopital's Rule?

L'Hopital's rule is unfortunately not allowed in this particular problem.