Answer
$$4$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {\frac{\pi }{2},0,\frac{\pi }{2}} \right)} 4\cos y\sin \sqrt {xz} \cr
& {\text{evaluate using the theorem 12}}{\text{.1}}{\text{, }} \cr
& {\text{substitute }}\frac{\pi }{2}{\text{ for }}x,{\text{0 for }}y{\text{ and }}\frac{\pi }{2}{\text{ for }}z{\text{ into }}f\left( {x,y,z} \right) = 4\cos y\sin \sqrt {xz} \cr
& = 4\cos \left( 0 \right)\sin \sqrt {\left( {\frac{\pi }{2}} \right)\left( {\frac{\pi }{2}} \right)} \cr
& {\text{simplifying}} \cr
& = 4\left( 1 \right)\left( 1 \right) \cr
& = 4 \cr} $$
