Chapter 13 - Functions of Several Variables - Review Exercises - Page 960: 11

$$\frac{1}{2}$$

$$\eqalign{ & \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {1,1} \right)} \frac{{xy}}{{{x^2} + {y^2}}} \cr & {\text{Evaluate by direct substitution, 1 for }}x{\text{ and }}1{\text{ for }}y \cr & = \frac{{\left( 1 \right)\left( 1 \right)}}{{{{\left( 1 \right)}^2} + {{\left( 1 \right)}^2}}} \cr & = \frac{1}{2} \cr} $$