Question

lim(x,y) -->(2,0) of (xy-2y)/(x^2+y^2-4x+4). Find the limit if it exists, or show that it doesn't exist.

Answer 1

Factor the function first:\[{xy-2y\over x^2+y^2-4x+4}={y(x-2)\over(x-2)^2+y^2}\]
Then make the substitution u=x-2, so that (u,y)->(0,0) as (x,y)->(2,0):
\[\lim_{(x,y)\to(2,0)}{y(x-2)\over(x-2)^2+y^2}=\lim_{(u,y)\to(0,0)}{uy\over u^2+y^2}\]
Now evaluate this last limit on two paths: along u=0, and along u=y.