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 $b$:

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

if and only if:

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 $\epsilon$.

Examples

Limit of $x^2 + y^2$ at $0$

Let $f$ be the real function of $2$ variables defined as:

$\map f {x, y} = x^2 + y^2$

Then:

$\ds \lim_{\substack {x \mathop \to 0 \\ y \mathop \to 0} } \map f {x, y} = 0$

where $\lim$ denotes the limit of $f$.

Limit of $\dfrac {x - y} {x + y}$ at $0$

Let $f$ be the real function of $2$ variables defined as:

$\map f {x, y} = \begin {cases} \dfrac {x - y} {x + y} & : x \ne -y \\ 1 & : x = -y \end {cases}$

Then:

$\ds \lim_{\substack {x \mathop \to 0 \\ y \mathop \to 0} } \map f {x, y}$

does not exist.

Also see

• Definition:Limit of Real Function

• Results about limits of real-valued functions can be found here.

Sources

• 1961: David V. Widder: Advanced Calculus (2nd ed.) ... (previous) ... (next): $1$ Partial Differentiation: $\S 3$. Functions of Several Variables: $3.1$ Limits and Continuity