Answer
$$\frac{1}{2}$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1,1} \right)} \frac{{xy}}{{x + y}} \cr
& {\text{use the law 5 }}\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {a,b} \right)} \frac{{f\left( {x,y} \right)}}{{g\left( {x,y} \right)}}.\,\,\,{\text{theorem 12}}{\text{.2 }}\left( {{\text{see page 887}}} \right){\text{. then}} \cr
& = \frac{{\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1,1} \right)} xy}}{{\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1,1} \right)} \left( {x + y} \right)}} \cr
& {\text{evaluate using the theorem 12}}{\text{.1}} \cr
& = \frac{{\left( 1 \right)\left( 1 \right)}}{{1 + 1}} \cr
& {\text{simplifying}} \cr
& = \frac{1}{2} \cr} $$
