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,z} \right) \to \left( {0,0,0} \right)} \frac{{x + y + z}}{{{x^2} + {y^2} + {z^2}}} = \mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {0,0,0} \right)} \frac{x}{{{x^2}}} = \mathop {\lim }\limits_{\left( {x,y,z} \right) \to \left( {0,0,0} \right)} \frac{1}{x} = \infty $
Therefore, the limit does not exist.
By symmetry we get the same results if we approach the origin along the $y$-axis or the $z$-axis.
