\begin{definition}[Definition:Limit of Real Function of 2 Variables]
Let $f: \R \times \R \to \R$ be a real function of $2$ variables.
Then $\map f {x, y}$ '''tends to the limit $A$ as $x$ approaches $a$ and $y$ approaches $b$''':
:$\ds \lim_{\substack {x \mathop \to a \\ y \mathop \to b} } \map f {x, y} = A$
{{iff}}:
:for each positive real number $\epsilon$ there exists another positive real number $\delta$ such that:
::whenever $\size {x - a} < \delta$ and $\size {y - b} < \delta$ such that $0 < \paren {x - a}^2 + \paren {y - b}^2$, we have:
:::$\size {\map f {x, y} - A} < \epsilon$
That is, when $\tuple {x, y}$ is at any point inside a square somewhere (except for its center) of side $2 \delta$ with center $\tuple {a, b}$, $\map f {x, y}$ is different from $A$ by less than $\epsilon$.
\end{definition}