Answer
$$\frac{\pi }{2}$$
Work Step by Step
$$\eqalign{
& \mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {0,0,0} \right)} {\tan ^{ - 1}}\left[ {\frac{1}{{{x^2} + {y^2} + {z^2}}}} \right] \cr
& {\text{Using the limit laws }} \cr
& = {\tan ^{ - 1}}\left[ {\frac{1}{{\mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {0,0,0} \right)} {x^2} + \mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {0,0,0} \right)} {y^2} + \mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {0,0,0} \right)} {z^2}}}} \right] \cr
& {\text{Evaluating the limit}}{\text{, substitute 0 for }}x,\,\,y{\text{ and }}z \cr
& = {\tan ^{ - 1}}\left[ {\frac{1}{{{{\left( 0 \right)}^2} + {{\left( 0 \right)}^2} + {{\left( 0 \right)}^2}}}} \right] \cr
& {\text{simplifying}} \cr
& = {\tan ^{ - 1}}\left[ {\frac{1}{0}} \right] \cr
& = {\tan ^{ - 1}}\left( { + \infty } \right) \cr
& = \frac{\pi }{2} \cr} $$
