What does the limit of the norm of a vector-valued function mean?!

Question

Given the vector-valued function $\mathbf r(t)=\left\langle t,t^2,t^3 \right\rangle$, what does the following expression mean?$$\lim_{t\to a}\left\Vert \mathbf r(t)\right\Vert$$If it really means the norm of the vector that $\mathbf r(t)$ will approach as $t\to a$, shouldn't it be written as this?$$\left\Vert\lim_{t\to a} \mathbf r(t)\right\Vert$$

Answer 1

As it's written, the first expression does mean the value that the norm approaches. But since the norm is a continuous functions, this is equal to the norm of the limit of the vector. In other words, the two expressions are equal. (Assuming that both exist; the first limit may exist even if the second one doesn't; for example, if $\mathbf{r}(t)$ oscillates erratically as $t \to a$, but has the same length all the time.)