Hello everyone!

I need some help with solving limits with Taylor's, I didn't really get the whole process yet.

I need to solve this guy:

Alright, I didn't really get where I am supposed to "stop" writing polinomials, my teacher said that I should stop when I find the smallest degree factor, because that's the one which is "bossing around" when the limit approaches zero.

Okay, that's where I've gone so far (see attachment)

I don't get if I wrote too many, if I didn't write enough terms, if I did something wrong at all, or I am right and should keep on doing calcs. Could someone help me out please? Thanks a lot!