Definition:Limit of Vector-Valued Function/Definition 1

Definition
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.