It is part of a larger problem, but the only hangup I have had is computing this limit.
lim x->0 e^(-1/x^2)/x^3.
It's to show that the function f(x) = e^(-1/x^2) (when x is not 0) and 0 (when x is 0) is not equal to its Maclurin Series. I know that if I can show that the derivatives of f(x) are equal to zero when x=0 (a) then I can prove what the problem is asking. Showing why the derivatives equal zero has been a problem thus far though...
The Attempt at a Solution
I put it into Wolfram Alpha and know that it's equal to zero, but I don't know how to get it.
I know that L'Hopital's rule applies because it is 0/0
((2/x^3)*e^(-1/x^2))/(3x^2) = 2e^(-1/x^2)/3x^5
Doesn't this pattern keep repeating giving one 0/0 no matter how many times one differentiated? It will always have e^(-1/x^2) on the top and some power of x which will continue to grow on the bottom. How can this limit be computed? Thanks!