Answer
limit does not exist
Work Step by Step
We notice that if we directly substitute limits in the given function $f(x,y)=\frac{xy-y}{(x-1)^{2}+y^{2}}$
Then $f(0,0)=\frac{0}{0}$
To evaluate limit along x-axis; put $y=0$
$f(x,0)=\frac{xy-y}{(x-1)^{2}+y^{2}}=\frac{x.0-0}{(x-1)^{2}+y^{2}}=0$
To evaluate limit along y-axis; put $x=1$
$f(1,y)=\frac{xy-y}{(x-1)^{2}+y^{2}}=\frac{y-y}{(1-1)^{2}+y^{2}}=0$
However, the obtained identical limits along the axes do not show that the given limit is 0.
Then, approach (0,0) along another line, $y=x-1$ and $x\ne0$
$f(x,x-1)=\frac{x(x-1)-(x-1)}{(x-1)^{2}+(x-1)^{2}}=\frac{1}{2}$
For a limit to exist, all the paths must converge to the same point.
Hence, both the limits are different and follow different paths, therefore, the limit does not exist.
