Answer
See explanation
Work Step by Step
The question is:$\lim\limits_{(x, y) \to (0,0)} \frac{2xy}{x^2+3y^2}$ .
Condition (a.) x=y:
$\lim\limits_{(x, y) \to (0,0)} \frac{2xy}{x^2+3y^2} \overset{x=y}{=} \lim\limits_{y \to 0} \frac{2y^2}{4y^2} = \frac{1}{2}$.
Condition (b.) x=-y:
$\lim\limits_{(x, y) \to (0,0)} \frac{2xy}{x^2+3y^2} \overset{x=-y}{=} \lim\limits_{y \to 0} \frac{-2y^2}{4y^2} = -\frac{1}{2}$.
Along different paths they have different limit here, so the limit does not exist.
