Answer
The limit does not exist
Work Step by Step
Let's make the table:
\[
\begin{array}{|c|c|c|c|}
\hline
\textbf{x\y} & \textbf{-0.05} & \textbf{0.05}& \textbf{0.1} \\ \hline
\textbf{-0.05} & 0.6667 & -0.6667 & -0.5\\ \hline
\textbf{0.05} & -0.6667 & 0.6667 & 0.5 \\ \hline
\textbf{0.1} & -0.5 & 0.5 & 0.667\\ \hline
\end{array}
\]
The function doesn't seem to have limit at origin.
Let's consider the line $y=x$:
$\lim_{(x, y)\rightarrow(0,0)}\frac{2xy}{x^2+2y^2}=\lim_{(x, x)\rightarrow(0,0)}\frac{2x^2}{x^2+2x^2}=\frac{2}{3}$
Now let's consider the line $y=-x$:
$\lim_{(x, y)\rightarrow(0,0)}\frac{2xy}{x^2+2y^2}=\lim_{(x, -x)\rightarrow(0,0)}\frac{-2x^2}{x^2+2x^2}=-\frac{2}{3}$
As the limits are distinct, the function has no limit at the origin.
