Why is lim(x->0) 1/(x^n), where n is even = infinity and where it's odd undefined?

For example,

lim(x->0) 1/x^2 = 0 ... why?

1/0 = indeterminant form... Derivative of 1 = 0...derivative of x^2 = 2x...

0/0 ... 0/2... = 0. Where does infinity come into play?