Definition:Limit of Vector-Valued Function

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Definition 1


$\mathbf r: t \mapsto \begin {bmatrix} \map {f_1} t \\ \map {f_2} t \\ \vdots \\ \map {f_n} t \end {bmatrix}$

be a vector-valued function.

The limit of $\mathbf r$ as $t$ approaches $c$ is defined as follows:

\(\ds \lim_{t \mathop \to c} \map {\mathbf r} t\) \(:=\) \(\ds \begin {bmatrix} \ds \lim_{t \mathop \to c} \map {f_1} t \\ \ds \lim_{t \mathop \to c} \map {f_2} t \\ \vdots \\ \ds \lim_{t \mathop \to c} \map {f_n} t \end {bmatrix}\)

where each $\lim$ on the right hand side is a limit of a real function.

The limit is defined to exist if and only if all the respective limits of the component functions exist.

Definition 2

Let $\mathbf r : \R \to \R^n$ be a vector-valued function.

We say that:

$\ds \lim_{t \mathop \to c} \map {\mathbf r} t = \mathbf L$

if and only if:

$\forall \epsilon > 0: \exists \delta > 0: 0 < \size {t - c} < \delta \implies \norm {\map {\mathbf r} t - \mathbf L} < \epsilon$

where $\norm {\, \cdot \,}$ denotes the Euclidean norm on $\R^n$.

Also see