Answer
0
Work Step by Step
We find:
\begin{array}{l}
x=\rho \sin \phi \cos \theta, \quad y=\rho \sin \phi \sin \theta, \quad z=\rho \cos \phi \\
\left|\frac{x y z}{x^{2}+y^{2}+z^{2}}\right|=\left|\frac{\rho^{3} \sin ^{2} \phi \cos \phi \sin \theta \cos \theta}{\rho^{2}}\right| \leq \rho \\
\lim _{\rho \rightarrow 0+} \rho=0 \quad \Rightarrow \quad \lim _{(x, y, z) \rightarrow(0,0,0)} \frac{x y z}{x^{2}+y^{2}+z^{2}}=0
\end{array}
