Definition of Limit in Multivariable Calculus Let $f(x,y)$ be a function defined in some disk that is centered at $(x_0,y_0)$. Suppose that $L$ is some real number. Then in this case, what does it mean that:
$$\lim_{(x,y)\to (x_0,y_0)} f(x,y)=L$$
So basically I want the definition of limit in this case. I want it in a way that doesn't use non-mathematical words such as "approaches", "close", etc. Instead I want it to be in a mathematical context that is understandable to someone taking multivariable calculus for the first time with only the knowledge from multivariable calculus and calculus 1 and 2.
 A: The definition of the limit is as follows:
$\lim_{(x,y)\to (x_0,y_0)}f(x,y)=L\,$ if and only if for all $\epsilon>0$ there exists a $\delta >0$ such that 
$$|f(x,y)-L|<\epsilon$$
whenever $0<\sqrt{(x-x_0)^2+(y-y_0)^2}<\delta$.

NOTE:  Alternatively, 
$\lim_{(x,y)\to (x_0,y_0)}f(x,y)=L\,$ if and only if for all $\epsilon>0$ there exists a $\delta >0$ such that 
$$|f(x,y)-L|<\epsilon$$
whenever $0<|x-x_0|<\delta$ and $0<|y-y_0|<\delta$.
A: The natural extension of the classical $\epsilon-\delta$ definition of limit to a function of two variable is:

The function $f(x,y)$ has limit $l$ as $(x,y)\rightarrow (x_0,y_0)$ if
  for every $\epsilon>0$ there exists $\delta>0$ such that
  $|f(x,y)-l|<\epsilon$ whenever $0<\sqrt{(x-x_0)^2+(y-y_0)^2}<\delta$.

Note that this means that we have to get the same limit no matter
from which direction we approach $(x_0,y_0)$.
A: If you're familiar with the $\epsilon-\delta$ definition of a single variable limit, it's very easy to adapt to the higher dimensional case. If we write $|x-c|$ as $d_\mathbb{R}(x,c)$, the definition of a one dimensional limit becomes: $\lim_{x\to c}f(x) = L$ iff $\forall \epsilon > 0.\exists\delta>0:d_\mathbb{R}(x,c)<\delta\implies d_\mathbb{R}(f(x),L) < \epsilon$. Informally, this simply says that we can force $f(x)$ to be arbitrarily close to $L$ by forcing $x$ to be arbitrarily close to $c$. The notation $d_\mathbb{R}$ makes especially clear that we are indeed talking about distances between real numbers.
This definition can be generalized to any set in which there is a notion of distance between points (such sets are called metric spaces). There is a notion of distance in euclidean space $\mathbb{R}^n$ given by the pythagorean theorem: $d_{\mathbb{R}^n}((a_1,...,a_n),(b_1,...,b_n)) = \sqrt{(a_1-b_1)^2+...+(a_n-b_n)^2}$. In two dimensional space, $\mathbb{R}^2$,this reduces to $d_{\mathbb{R}^2}((x_1,y_1),(x_2,y_2)) = \sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$.
We can simply replace the distance in $\mathbb{R}$ with the distance in $\mathbb{R}^2$ to obtain a definition of two-dimensional limits: $\lim_{(x,y)\to c(x_0,y_0)}f(x,y) = L$ iff $\forall \epsilon > 0.\exists\delta>0:d_\mathbb{R^2}((x,y),(x_0,y_0))<\delta\implies d_\mathbb{R}(f(x,y),L) < \epsilon$. 
Informally, this says we can force $f(x)$ to be arbitrarily close to $L$ by forcing $(x,y)$ to be inside some circle of arbitrarily small radius about $(x_0,y_0)$.
Notice that we could also change the target space, i.e. we can define continuity of a function from $\mathbb{R^2}$ to $\mathbb{R^3}$ by changing the distance function we use in the $d(f(x),L)$ term; limits can be defined for any function from any one metric space to any other metric space.
