Answer
Does not exist.
Work Step by Step
Along the line $x=0$, the value of limit is zero.
Along $x=0$: $\lim\limits_{(x,y) \to (0,0)}\frac{2xy}{x^2+2y^2}=0$
Along $y=x$: $\frac{2x^2}{x^2+2x^2}=\frac{2}{3}$
Since along two different lines the limit has two different values, it does not exist.
