Not sure how to find this limit algebraically.
However, you can evaluate it analytically.
Notice the domain is restricted to:
[tex]x^2 \ \textgreater \ 4[/tex]
This means the right sided limit cannot exist.
You can only evaluate the left side limit.
As x-> -2 from the left. The first term goes to positive infinity and 2nd term goes to negative infinity.
[tex]\infty - (-\infty) = \infty[/tex]
Therefore the limit is positive infinity.
