Definition:Limit of Vector-Valued Function/Definition 2

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

We say that:
 * $\ds \lim_{t \mathop \to c} \map {\mathbf r} t = \mathbf L$


 * $\forall \epsilon > 0: \exists \delta > 0: 0 < \size {t - c} < \delta \implies \norm {\map {\mathbf r} t - \mathbf L} < \epsilon$
 * $\forall \epsilon > 0: \exists \delta > 0: 0 < \size {t - c} < \delta \implies \norm {\map {\mathbf r} t - \mathbf L} < \epsilon$

where $\norm {\, \cdot \,}$ denotes the Euclidean norm on $\R^n$.

Also see

 * Equivalence of Definitions of Limit of Vector-Valued Function