Let $f:A \subset \mathbb{R} \mapsto \mathbb{R}^n $, $b\in \mathbb{R}^n $ and $a\in A´$
$$\text{lim}_{t \rightarrow a } f(t) = b \Leftrightarrow \text{lim}_{t \rightarrow a} \Vert f(t) \Vert = \Vert b \Vert$$
$\Rightarrow ]$ Its true because $\mid \Vert f(t) \Vert - \Vert b \Vert \mid \leq \Vert f(t)-b \Vert. \text{Choose } \delta = \epsilon.$
$\Leftarrow ]$ It seems to me that this doesn't need to happen, but I can't find a counterexample to prove it. I know it holds when the function is continuous because the norm is continuous and because the composition of continuous funtions is continuous. What about the limit only? Can we generalize it to any $(\Bbb{V},\Bbb{K},\Vert \cdot \Vert)$ normed vector space? In this particular case we have a similar equivalence: $$ \text{lim}_{t \rightarrow a} f(x) = b \Leftrightarrow \text{lim}_{t \rightarrow a} \Vert f(t) - b \Vert = 0_{\Bbb{K}}$$