Proof of derivatives through induction

Hey guys,

Have a problem that was set as exam revision and was told that there'd be a question similar on the exam, so I'm very keen on understanding how to do this problem well.

The problem asks:

Let f(x) = {e^{-1/x^2}, if x =/= 0

{0, if x = 0

Prove that f^{(n)}(0) = 0 for positive integer values of n.

I've figured that induction would be a good way to go about solving this problem obviously, although I'm not too sure how to prove it for any case.

Any help would be greatly appreciated :)

Thanks in advance,

Mark

Re: Proof of derivatives through induction

You need the fact that $\displaystyle \lim_{t\to\infty}e^{-t^2}t^n=0$ for all n.

Prove by induction on n that there exists a polynomial $\displaystyle p_n$ such that $\displaystyle f^{(n)}(x)=e^{-1/x^2}p_n(1/x)$ for $\displaystyle x\ne 0$ and $\displaystyle f^{(n)}(0) = 0$. The latter part for n + 1 is proved by the definition of the derivative using the induction hypothesis.