Answer
Does not exist
Work Step by Step
Given $$ \lim _{(x, y) \rightarrow(0,0)} \frac{x y}{3 x^{2}+2 y^{2}}$$
Consider the line $y=mx$ that passes through $(0,0)$:
\begin{align*}
\lim _{(x, y) \rightarrow(0,0)} \frac{x y}{3 x^{2}+2 y^{2}}&=\lim _{x,\rightarrow0} \frac{mx^2}{3 x^{2}+2m^2 x^{2}}\\
&=\lim _{x,\rightarrow0} \frac{m}{3 +2m^2 }\\
&=\frac{m}{3 +2m^2 }
\end{align*}
Since the limit depends on $m$, then $ \lim _{(x, y) \rightarrow(0,0)} \dfrac{x y}{3 x^{2}+2 y^{2}}$ does not exist.
