Question

Find the limit or verify that the limit does not exist: lim(x,y)--->(0,2) f(x,y)=(x+y-2)/(x^2+y^2+2xy-4)

Answer 1

\[\lim_{(x,y)\to(0,2)}\frac{x+y-2}{x^2+y^2+2xy-4}\]
First thing that stands out to me is that
\[x^2+2xy+y^2=(x+y)^2\]
so you have
\[\lim_{(x,y)\to(0,2)}\frac{(x+y)-2}{(x+y)^2-4}\]
i.e. a difference of squares in the denominator, which can be factored:
\[\lim_{(x,y)\to(0,2)}\frac{(x+y)-2}{((x+y)-2)((x+y)+2)}\\
\lim_{(x,y)\to(0,2)}\frac{1}{x+y+2}\]
No discontinuities/indeterminate forms to worry about, so you can evaluate directly.