# Trouble with $\epsilon-\delta$ in multivariable limits

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.

• Hint: $|(x,y)-(1,2)| < \delta \implies |x-1| < \delta \text{ and } |y-2| < \delta$. Sep 24, 2016 at 0:00

• There are lots of different notions of distance in the plane. The one you are used to is the euclidean norm, which looks like a ball or disc. However, you can use rectangles, i.e. of the form $|x-a|+|y-b|$. Draw a picture to convince yourself that you can fit a rectangle inside any circle, and then use this norm instead since it is much easier Sep 23, 2016 at 21:58