I've been having trouble with $\epsilon$-$\delta$ proofs of multivariable limits, even simple ones like $$\lim_{ \begin{pmatrix} x \\ y \\ \end{pmatrix} \to \begin{pmatrix} 1 \\ 2 \\ \end{pmatrix} }{x^2\over x+y}={1\over3}$$ I can't move (much) beyond stating the definition $$(\forall\begin{pmatrix} x\\ y\\ \end{pmatrix})(\forall\epsilon>0)(\exists\delta>0) (\lvert \begin{pmatrix} x \\ y\\ \end{pmatrix} - \begin{pmatrix} 1\\ 2\\ \end{pmatrix} \rvert<\delta \longrightarrow \lvert{x^2\over x+y}-{1\over3}\rvert<\epsilon) $$ Are there any tips or resources on how to prove multivariable limits?
Also, I know (1,2) is in the domain of the function, which is how I found 1/3. My problem is with $\epsilon-\delta$ proofs in more than one variable.
It's my first question here so I apologise if I made anything I shouldn't have.
A general tip: inside every ball is a rectangle/every norm on the real plane is the same up to a constant you don't care about. This avoids dealing with messy squares and square roots.
Other than that, start with your epsilon bound and modify it until you have the bound in your domain, then work backwards