Originally Posted by

**arbolis** Say whether or not the limit of $\displaystyle f(x,y)$ when $\displaystyle (x,y)$ tends to $\displaystyle (0,0)$ exists.

$\displaystyle f(x,y)=\frac{xy}{x^2-y^2}-2y$.

My attempt : $\displaystyle \lim _{(x,y) \to (0,0)}f(x,y)=\lim _{x\to 0} f(x,0)=0$.

$\displaystyle \lim _{(x,y) \to (0,0)}f(x,y)=\lim _{x\to 0} f(x,x^2)=-\infty$. As both limits are different, the limit doesn't exist.

However I'm unsure I didn't make an error by taking $\displaystyle y=x^2$ for the second limit. Since when $\displaystyle x=1$, this doesn't even make sense to take such an $\displaystyle y$.

The domain of $\displaystyle f$ is any $\displaystyle (x,y) \in \mathbb{R}^2$ such that $\displaystyle x\neq \pm y$.