I've found your problem really delicious to deal with. I wanted to make it clear as well.
\lim_{x\rightarrow 0}\left ( \frac{Ae^{A/x^2}+Be^{B/x^2}}{e^{A/x^2}+e^{B/x^2}}\right )
Let \phi = 1/x^2
When x\rightarrow 0, \phi\rightarrow \infty
Substitute \phi
\lim_{\phi\rightarrow \infty}\left (...