Answer
Limit does not exist
Work Step by Step
We need to find the limit for $\lim\limits_{(x,y) \to (0,0)} \dfrac{y}{x^2-y}$
Let us consider $y=a x^2$; $a \ne 1$
Then $\lim\limits_{(x,y) \to (0,0)} \dfrac{y}{x^2-y}=\lim\limits_{(x,ax^2) \to (0,0)} \dfrac{ax^2}{x^2-ax^2}=\lim\limits_{(x,ax^2) \to (0,0)} \dfrac{ax^2}{x^2(1-a)}$
Thus, $=\dfrac{a}{1-a}$
Thus, the limit does not exist as it has a different limit for different values of $a$.
