> if
> f(x)=
> {0,                      if x=0  
> p(x)/q(x) *(e^{-1/(x^2)} if x neq 0

> where p and q are poly's, defined off the nonzero roots of q.  Prove lim
> x->0 f(x)=0 and f(x) is infinetely differentiable.

> My solution starts like this: lim p(x)/q(x) *(e^{-1/(x^2)}=lim p(x)/q(x) *
> lim e^{-1/(x^2)} if both limits exist and are finite.   By L'hopitals rule
> lim p(x)/q(x)=lim p'(x)/q'(x)... and so on.  If deg(p(x))>deg(q(x)) then
> this chain equals lim p^(n)(x)/1 which is just some constant. and similiary
> if deg(p(x))<deg(q(x)).  Thus since lim e^{-1/(x^2)=0 we have it.

> But for showing its C^inf im not sure what to do...