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