Definition:Limit of Vector-Valued Function/Definition 2

Definition
Let $\mathbf r : \R \to \R^n$ be a vector-valued function.

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


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

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