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