In a proof to show that $D(L\circ f)_a = L \circ Df_a$, where $f: U \subset E \rightarrow F$ and $L \in L_c(F,G)$.
They take the following limit to show that the frechet derivative exists,
$\lim_{h\rightarrow 0} \frac{(F\circ f)(a+h) - (L \circ f)(a) - (L\circ Df_a)(h)}{\lVert h \rVert}$
My question: normally, in the definition of the frechet derivative the numerator is contained within a norm. Here, this is not the case. My guess is that we are allowed to do this, since the norm is continuous, and $\lVert x \rVert = 0$ iff $x = 0$. But aren't we always allowed to do this? Furthermore, if we are always allowed to do this, then why does the frechet derivative include a norm in its numerator?