For $f(x,y) = \frac{xy-1}{x^2 y^2-1}$, what is the limit as $(x,y)$ goes to $(1,1)$? 
For $f(x,y) = \frac{xy-1}{x^2 y^2-1}$, what is the limit as $(x,y)$ goes to $(1,1)$?

Since the denominator can be factored into $(xy-1)(xy+1)$ and then the $(xy-1)$'s in both the numerator and denominator can be cancelled, we are left with $f(x,y) = \frac{1}{xy+1}$ and substituting $x=1$ and $y=1$, the limit becomes $\frac{1}{2}$. Is this solution right? We had this question on our test todayin Multivariable Caclulus, and some people put $1/2$ while others put $DNE$. Who is right?

Consider $X=x-1$ and $Y=y-1$, then we have 
$$
\begin{array}{c}
\lim_{(x,y)\to (1,1)}\frac{xy-1}{x^2 y^2-1}=
\lim_{(X,Y)\to (0,0)}\frac{XY-Y-X}{ \left( XY-Y-X+2 \right)  \left( XY-Y-X \right)}
=\lim_{(X,Y)\to (0,0)}\frac{1}{ \left( XY-Y-X+2 \right)}=\frac{1}{2}
\end{array}
$$
