Definition:Limit of Vector-Valued Function

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Definition 1


$\mathbf r: t \mapsto \begin{bmatrix} f_1\left({t}\right) \\ f_2\left({t}\right) \\ \vdots \\ f_n\left({t}\right) \end{bmatrix}$

be a vector-valued function.

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

\(\displaystyle \lim_{t \to c} \ \mathbf r \left({t}\right)\) \(:=\) \(\displaystyle \begin{bmatrix} \lim_{t \to c} \ f_1\left({t}\right) \\ \lim_{t \to c} \ f_2\left({t}\right) \\ \vdots \\ \lim_{t \to c} \ f_n\left({t}\right)\end{bmatrix}\)

where each $\lim$ on the RHS is a limit of a real function.

The limit is defined to exist precisely when 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:

$\displaystyle \lim_{t \to c} \ \mathbf r\left({t}\right) = \mathbf L$

if and only if:

$\forall \epsilon > 0: \exists \delta > 0: 0 < \left\vert {t - c} \right\vert < \delta \implies \left\Vert \mathbf r \left({t}\right) - \mathbf L \right\Vert < \epsilon$

where $\left\Vert {\cdot} \right\Vert$ denotes the Euclidean norm on $\R^n$.

Equivalence of Definitions

That the definitions above are in fact equivalent is shown on Equivalence of Definitions of Limit of Vector-Valued Function.

Also see