Multivariable limit properties

Question

I'd like to prove some basic (non-trivial) limit properties for multivariable functions to myself using epsilon-delta bounding. In particular, I would like to show the basic "sum of limits" property from single variable calculus, but generalized to multivariable functions. Starting off with the simplest case of a function of 2 variables (I'll generalize after that, but it might get ugly?) - how do I begin to proceed here? I'm having trouble with how what I thought of as "distances" (absolute value) in single variable calculus translate to distances in, say, Euclidean 2-space. Is this as simple as "given $\epsilon > 0$ ... $|f(x,y) - L| < \epsilon$ and $|g(x,y) - M| < \epsilon$ whenever $\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} < \delta$"? I only have familiarity with single variable epsilon-delta.

A precise definition (which is seemingly hard to find on the internet) of a multivariable limit in terms of epsilon-delta would be helpful for me here as well.

Answer 1

$\lim_{(x,y) \to (a,b)} f(x,y)=L$ if, given $\epsilon >0$, there exists $\delta >0$ such that $0<\sqrt {(x-a)^{2}+(y-b)^{2}} <\delta$ implies $|f(x,y)-L| <\epsilon$.

Answer 2

In $R^n$ we have the norm $||x||_d=\sqrt{x_1^2+...+x_n^2}$ and in $R$ we have as norm the usual absolute value.

For $f: R^n \to R$

$\lim_{(x_1,x_2,...,x_n) \to (a_1,...,a_n)}f(x_1,...,x_n)=L \text{ if } \forall \epsilon>0 \exists \delta >0$ such that $|f(x_1,...,x_n)-L|<\epsilon,\forall x: ||(x_1,...,x_n)-(a_1,...,a_n)||<\delta$