Evaluate:
I understand the answer is negative infinity but only by using the table method. Is there an analytic method possible?
Perhaps you are thinking about the L'Hopital's rule. In such a case, take into account that
$\displaystyle \displaystyle\lim_{x\to -2^-}\;x^2=4\;,\quad\displaystyle\lim_{x\to -2^-}(x+2)=0$
so, the corresponding hypothesis are not satisfied.
Edited: Sorry, I didn't see DrSteve's post.