Answer
The limit does not exist.
Work Step by Step
Let us approach the origin along the $x$-axis. So, $y=z=0$. The limit becomes
$\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{{x^2} - {y^2} + {z^2}}}{{{x^2} + {y^2} + {z^2}}} = \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{{x^2}}}{{{x^2}}} = 1$
If we approach the origin along the $y$-axis, we have $x=z=0$. The limit becomes
$\mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{{x^2} - {y^2} + {z^2}}}{{{x^2} + {y^2} + {z^2}}} = \mathop {\lim }\limits_{\left( {x,y} \right) \to \left( {0,0} \right)} \frac{{ - {y^2}}}{{{y^2}}} = - 1$
Since we get different limits when $\left( {0,0} \right)$ is approached along different paths, the limit does not exist.
