I'm having some issues with $\varepsilon$-$\delta$ proofs of limits with more than one variable. I understand the $\varepsilon$-$\delta$ definition of a limit, but I don't know how to deal with multiple variables.
Here's a simple example: prove the following limit, if it exists, using the epsilon-delta definition: $$ \lim_{(x,y)\to (1,2)}\frac{x^2}{x+y} = a \iff \forall(\varepsilon>0)\, \exists(\delta) \left[ |(x,y)-(1,2)|<\delta \implies \left|\frac{x^2}{x+y}-a\right|<\varepsilon \right] $$
Obviously this limit is $1/3$, but how do I prove it?