Definition:Limit of Real Function of 2 Variables

Definition
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 $b$:
 * $\ds \lim_{\substack {x \mathop \to a \\ y \mathop \to b} } \map f {x, y} = A$


 * 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:
 * $\size {\map f {x, y} - A} < \epsilon$
 * $\size {\map f {x, y} - A} < \epsilon$

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 $\epsilon$.

Also see

 * Definition:Limit of Real Function