Answer
$$0$$
Work Step by Step
Given $$\lim _{(x, y) \rightarrow(0,0)} \frac{x^{3} y^{2}+x^{2} y^{3}}{x^{4}+y^{4}}$$
Choose two lines that pass through $(0,0) $
\begin{align*}
\frac{y}{x}&=m
\end{align*}
Let $y=x$; then
\begin{align*}
\lim _{(x, y) \rightarrow(0,0)} \frac{x^{3} y^{2}+x^{2} y^{3}}{x^{4}+y^{4}}&=\lim _{x \rightarrow 0} \frac{x^{5} +x^{5} }{x^{4}+x^{4}}\\
&=0
\end{align*}
Let $y=2x$; then
\begin{align*}
\lim _{(x, y) \rightarrow(0,0)} \frac{x^{3} y^{2}+x^{2} y^{3}}{x^{4}+y^{4}}&=\lim _{x \rightarrow 0} \frac{4x^{5} +8x^{5} }{x^{4}+16x^{4}}\\
&=0
\end{align*}
Hence
$$ \lim _{(x, y) \rightarrow(0,0)} \frac{x^{3} y^{2}+x^{2} y^{3}}{x^{4}+y^{4}}=0$$
