Prove: All Derivatives of f at 0 = 0 if Lim f(x)/x^n = 0 as x --> 0

[stukbv]

Homework Statement

if f is infinitely continuously differentiable and f(0) = 0 then prove that all derivatives of f at 0 are 0 iff lim f(x)/x^n = 0 as x --> 0

Homework Equations

The Attempt at a Solution

I didnt know whether to use induction on this,
I tried a base case so said that f'(0)=0 iff lim (f(x)/x) = 0 as x--> 0
But then it gets messy..
Think i might be on the wrong lines.

Thanks a lot

 

[micromass]

Yes, induction should be the way to go here. Did you already prove the base case? It shouldn't be too hard...

 

[ideasrule]

I don't think induction is the best approach. Try using the Taylor expansion of f instead.

EDIT: never mind, induction also works.

 

[micromass]

I don't think induction is the best approach. Try using the Taylor expansion of f instead.

EDIT: never mind, induction also works.

If you do Taylor expansion then you necessarily need to do induction. Note that f doesn't necessarily equal it's Taylor series!

 

[stukbv]

Yeah i can do the base case and that's all proved etc but then i get stuck

 

[micromass]

For the induction hypothesis, try to calculate

$$\frac{f(x)}{x^n}$$

by taking the Taylor expansion at 0. This will help you to evaluate the limit.