Answer
$${\text{The limit does not exist}}{\text{.}}$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1,1} \right)} \frac{{xy}}{{{x^2} - {y^2}}} \cr
& {\text{Evaluate by direct substitution, 1 for }}x{\text{ and }}1{\text{ for }}y \cr
& = \frac{{\left( 1 \right)\left( 1 \right)}}{{{{\left( 1 \right)}^2} - {{\left( 1 \right)}^2}}} \cr
& = \frac{1}{0} \cr
& {\text{Therefore,}} \cr
& {\text{The limit does not exist}}{\text{.}} \cr} $$
