Definition:Limit of Vector-Valued Function

Definition 1
Let:
 * $\mathbf r:t \mapsto \begin{bmatrix} f_1\left({t}\right) \\ f_2\left({t}\right) \\ \vdots \\ f_n\left({t}\right) \end{bmatrix}$

be a vector-valued function.

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

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

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

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

Also see

 * Limit of a Function (Metric Space)