## Thomas' Calculus 13th Edition

limit $l$ does not exist
Consider $l=\lim\limits_{(x,y) \to (0,0)} \dfrac{x^2+y^2}{xy}$ Suppose $y=p x$; $p \ne 0$ Now, $l=\lim\limits_{(x,px) \to (0,0)} \dfrac{x^2+p^2 x^2}{px^2}$ Hence, $l=\lim\limits_{(x,px) \to (0,0)} \dfrac{1+p^2}{p}=\dfrac{1+p^2}{p}$ Hence, the limit $l$ does not exist as we can see that $l$ will have different limits for different values of $p$.