If we have the function $f: (0,\infty) \rightarrow\ R$, $f(x) = e^{\frac{-1}{x^2}} $
Show that for $n≥0$, there exists a polynomial function $p_{(n)}(t)$ such that $f^{(n)} (x) =p_{(n)}(\frac{1}{x})f(x)$ for all $x>0$
Given any polynomial function $p(t)$, show that $\lim_{x\to0} p(\frac{1}{x})f(x)=0 $
Show that f is infinitely often differentiable and that the Taylor series of $f$ at $0$ is the zero series $0+0x+0x^2+...$
I think I need to use induction to do this, I already know that $\lim_{x\to\infty} \frac{x^n}{e^{x^2}}=0 $ for $n≥0$ Any help is appreciated.
