Answer
$$\frac{\pi }{4}$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {5,2, - 3} \right)} {\tan ^{ - 1}}\left( {\frac{{x + {y^2}}}{{{z^2}}}} \right) \cr
& {\text{evaluate using the theorem 12}}{\text{.1}}{\text{, }} \cr
& {\text{substitute }}5{\text{ for }}x,{\text{ 2 for }}y{\text{ and }} - 3{\text{ for }}z{\text{ into }}f\left( {x,y,z} \right) = {\tan ^{ - 1}}\left( {\frac{{x + {y^2}}}{{{z^2}}}} \right) \cr
& = {\tan ^{ - 1}}\left( {\frac{{5 + {{\left( 2 \right)}^2}}}{{{{\left( { - 3} \right)}^2}}}} \right) \cr
& {\text{simplifying}} \cr
& = {\tan ^{ - 1}}\left( {\frac{9}{9}} \right) \cr
& = {\tan ^{ - 1}}\left( 1 \right) \cr
& = \frac{\pi }{4} \cr} $$
