Answer
Des not exist
Work Step by Step
We find the limit along the $x-$axis ($y=0$):
\begin{align*}
\lim _{(x, y) \rightarrow(0,0)} f(x, y)&=\lim _{x \rightarrow 0} f(x, 0)\\
&=\lim _{x \rightarrow 0} 3=3
\end{align*}
and along the $y-$axis $(x=0)$:
\begin{align*}
\lim _{(x, y) \rightarrow(0,0)} f(x, y)&=\lim _{y \rightarrow 0} f(0, y)\\
&=\lim _{y \rightarrow 0} 5=5
\end{align*}
Since the limit depends on the path, $ \lim _{(x, y) \rightarrow(0,0)} f(x, y)$ does not exist.
