I am supposed to find the limit as x -> 0 of (e^(x^2) - 1 - x^2) / (x sinx - x^2).

I do not know if my process or answer is at all correct, but here is what I have done:

I found the taylor series for e^(x^2) = 1 + x^2 + x^4/2! + x^6/3! ...

then I added in the rest of the numerator (-1 - x^2) to get:

x^4/2

for the denominator, I found the taylor series for sinx and then added and multiplied in the rest of the denominator to get:

-x^4/6 + x^6/5! -x^8/7! ....

taking the ratio of the x^4 terms give me 1/2 * -6/1 = -3

is this correct?

if so, is it just normal practice to ignore the rest of the terms in the series and just focus on the first one? Thanks!!