Definition:Limit of Vector-Valued Function/Definition 1

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Definition

Let:

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

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.


Also see


Sources