Answer
Limit does not exist
Work Step by Step
We have to prove that limit of the function does not exist along the axes
(a)
\[
\lim _{(x, y) \rightarrow(0,0)} \frac{3}{x^{2}+2 y^{2}}
\]
So along $x=0$
\[
\begin{array}{l}
=\lim _{y \rightarrow 0} \frac{3}{(0)^{2}+2 y^{2}} \\
=\lim _{y \rightarrow 0} \frac{3}{2 y^{2}} \\
=\operatorname{not} \text { defined }
\end{array}
\]
We can see that limit does not exist.
(b)
\[
\lim _{(x, y) \rightarrow(0,0)} \frac{x+y}{2 x^{2}+y^{2}}
\]
So along $x=0$
\[
\begin{array}{l}
=\lim _{y \rightarrow 0} \frac{x+y}{2(0)^{2}+y^{2}} \\
=\lim _{y \rightarrow 0} \frac{1}{y}
=\text{undefined}
\end{array}
\]
We can see that the limit does not exist.
