Answer
Does not exist
Work Step by Step
Given $$\lim _{(x, y) \rightarrow(0,0)} \frac{x y+x y^{2}}{x^{2}+y^{2}}$$
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 y+x y^{2}}{x^{2}+y^{2}}&=\lim _{x \rightarrow 0} \frac{x^2+x^{3}}{x^{2}+x^{2}}\\
&=\frac{1}{2}
\end{align*}
Let $y=2x$; then
\begin{align*}
\lim _{(x, y) \rightarrow(0,0)} \frac{x y+x y^{2}}{x^{2}+y^{2}}&=\lim _{x \rightarrow 0} \frac{2x ^2+4x ^{3}}{x^{2}+4x^{2}}\\
&=\frac{2}{5}
\end{align*}
Since the two limits are different, then the given limit does not exist.
