Limits of Parametric Curves#
In the case of vector-valued functions \(\gamma: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}^n\), the definition of a limit reduces to the following.
Definition: Limits of Parametric Curves
Let \(\gamma: \mathcal{D} \subseteq \mathbb{R} \to \mathbb{R}^n\) be a vector-valued functions let \(t_0\) be an accumulation point of \(\mathcal{D}\).
We say that \(\mathbf{L} \in \mathbb{R}^n\) is the limit of \(\gamma\) for \(t \to t_0\) if and only if for each \(\varepsilon \gt 0\), there exists some \(\delta \gt 0\) such that for all \(t \in I\) we have
\[ 0 \lt ||t - t_0|| \lt \delta \implies ||\gamma(t) - \mathbf{L}|| \lt \varepsilon \]