Chapter 12.3, Problem 84E

Proof of Limit Law 1 Use the formal definition of a limit to prove that $\underset{(x,y)\to (a,b)}{\lim }\left(f\left(x,y\right)+g\left(x,y\right)\right)=\underset{(x,y)\to (a,b)}{\lim }f\left(x,y\right)+\underset{(x,y)\to (a,b)}{\lim }g\left(x,y\right)$.