Limit of a function of several variables

Builds on Euclidean space Function of several variables Absolute value

Definition

Let $D \subseteq \mathbb{R}^n$ and let $f \colon D \to \mathbb{R}^m$ . Let $a \in \mathbb{R}^n$ be a limit point of $D$ : each open ball about $a$ contains a point of $D$ other than $a$ . Let $L \in \mathbb{R}^m$ . When this holds, $f(x)$ tends to $L$ as $x$ tends to $a$ , with the notation $\lim_{x \to a} f(x) = L$ . For each $\varepsilon > 0$ there is a $\delta > 0$ such that $\|f(x) - L\| < \varepsilon$ for each $x \in D$ with $0 < \|x - a\| < \delta$ . Here $\|\cdot\|$ is the Euclidean norm of Euclidean space.

Remarks

A limit is unique when it exists, because $a$ is a limit point of $D$ . The value $f(a)$ , and even whether $a$ is in $D$ , has no effect on the limit. A function into $\mathbb{R}^m$ tends to $L$ if and only if each component $f_i$ tends to $L_i$ . To show that a limit does not exist, it is enough to find two paths with different limits.

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Absolute value concept
Euclidean space object
Function concept
Function of several variables concept
Limit of a function of several variables definition

Tag	`0VMG8TA8VF`
URI	`m2://0VMG8TA8VF`
Kind	definition
Writing	W0 stub
MSC	26B05
Defines	Limit of a function of several variables
Uses	Euclidean space, Function of several variables, Absolute value

Limit of a function of several variables

Definition

Remarks

Reading path

Further reading

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