Answer
${e^{\frac{\pi }{2}}}$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {\pi ,1/2} \right)} {e^{xy}}\sin xy \cr
& {\text{Evaluating the limit by direct substitution}}{\text{.}} \cr
& {\text{Substitute }}\pi {\text{ for }}x{\text{ and }}\frac{1}{2}{\text{ for }}y \cr
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {\pi ,1/2} \right)} {e^{xy}}\sin xy = {e^{\left( \pi \right)\left( {\frac{1}{2}} \right)}}\sin \left[ {\left( \pi \right)\left( {\frac{1}{2}} \right)} \right] \cr
& {\text{Simplifying}} \cr
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {\pi ,1/2} \right)} {e^{xy}}\sin xy = {e^{\frac{\pi }{2}}}\sin \left( {\frac{\pi }{2}} \right) \cr
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {\pi ,1/2} \right)} {e^{xy}}\sin xy = {e^{\frac{\pi }{2}}} \cr} $$
