Given a function $f \colon I \subset \mathbb{R} \to \mathbb{R}^n$, that has a limit ${L}$. Does the limit $|{L}|$ exist for |f|?
If $f$ is continuous at $a$ and $g$ is continuous, then $g(f(x))$ is continuous at $a$. In this case, $g= \vert \cdot \vert$, which is continuous.