Finding limit using polar coordinates Polar coordinates are often useful in finding the limits in the case where $(x,y) $ approaches  (0,0). Is it possible to use polar coordinates to find 
$$\lim_{(x,y)\rightarrow (1,1)} \frac {xy-y-2x+2}{x-1} $$
This limit is -1  by direct substitution, can I use polar coordinates

For this case you would have to use $x=r\cos\theta+1,y=r\sin\theta+1$, giving
$$\lim_{r\to0}\frac{(r\sin\theta-1)r\cos\theta}{r\cos\theta}$$ where $\theta$ can vary arbitrarily. This indeed shows that the limit exists and is $-1$, but there is no benefit to switch to polar.

An alternative approach would be to use translated Cartesian coordinates, $u=x+1,v=y+1$, and
$$\lim_{(x,y)\to(-1,-1)}\frac{xy-y-2x+2}{x-1}=\lim_{(u,v)\to(0,0)}\frac{(v-1)u}u.$$
