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 $\epsilon >0$, there exists some $\delta >0$ such that for all $t\in I$ we have

$$0<||t-t_0||<\delta ⟹||\gamma (t)-\mathbf{L}||<\epsilon$$