Evaluate $\lim_{(x,y)\to(1,-1)}\frac{xy+1}{x^2-y^2}$

Question

$$\lim_{(x,y)\to(1,-1)}\frac{xy+1}{x^2-y^2}$$

I was going to try polar coordinates but I don't know if I should input $-1$ or $1$ into $r$. I also tried L'Hopital's rule but don't know whether I should get the derivative with respect to $x$ or $y$.

Answer 1

$y=-1:$

$\dfrac{1-x}{(x-1)(x+1)}= -\dfrac{1}{x+1}.$

$\lim_{ x \rightarrow 1} -\dfrac{1}{x+1}=-1/2$.

$x=1:$

$\dfrac{y+1}{1-y^2}= \dfrac{y+1}{(1-y)(1+y)}= \dfrac{1}{1-y}.$

$\lim_{y \rightarrow -1} \dfrac{1}{1-y}=1/2$.

The limit does not exist .