\begin{definition}[Definition:Limit of Vector-Valued Function/Definition 1]
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:
{{begin-eqn}}
{{eqn | l = \lim_{t \mathop \to c} \map {\mathbf r} t
      | o = :=
      | r = \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}
}}
{{end-eqn}}
where each $\lim$ on the {{RHS}} is a limit of a real function.
The limit is defined to exist {{iff}} all the respective limits of the component functions exist.
\end{definition}