Limits of '0/0' functions

How can find a limit, or prove it doesn't exist, for the following function:

When I differentiate, I get another function with a similar problem, and it just gets more complicated, so I can assume l'Hôpital is not what I need to use here.

I tried to simplify the function by multiplying it by , and then I got the function: but the changed the function and I guess I can't use it (moreover, it shows the limits are and , and in the original function I found that it doesn't go straight to this point, but 'gets crazy' and reaches infinity when it gets closer to 0 (from both sides).