Definition:Limit of Real Function of 2 Variables
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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:
- $\size {\map f {x, y} - A} < \epsilon$
- whenever $\size {x - a} < \delta$ and $\size {y - b} < \delta$ such that $0 < \paren {x - a}^2 + \paren {y - b}^2$, we have:
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
- 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