Why is it true that $o(\Vert t\vec u\Vert)=o(\vert t\vert)\cdot\Vert\vec u\Vert$? As far as I can rewrite, we have:
$$
\lim_{t\to0}\frac{o(\Vert t\vec u\Vert)}{\Vert t\vec u\Vert}=\lim_{t\to0}\frac{1}{\Vert\vec u\Vert}\cdot\frac{o(\Vert t\vec u\Vert)}{\vert t\vert}=0,
$$
so why don't we say $\begin{align}o(\Vert t\vec u\Vert)=\frac{1}{\Vert\vec u\Vert} o(\vert t\vert)\end{align}$?
I know it doesn't matter if we multiply by a constant $\Vert\vec u\Vert$, but I'm just confused why they multiply and don't divide by the constant. In the end it doesn't matter of course, because the constant disa…