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