Let $\mathbb{K}$ represent either $\mathbb{R}$ or $\mathbb{C}$ and let $E$ denote a normed vector space. Then, a map $f:X \subset \mathbb{K} \rightarrow E$ is said to be approximately linear at a point $a \in X$ if if there exists an affine function $g:\mathbb{K} \rightarrow E$ such that $f(a) = g(a)$ and
$\lim_{x \to a} \frac{\|f(x)-g(x)\|}{|x-a|} = 0$
My question is, is the presence of the norm and absolute value in the above definition necessary? The expression $x - a$ is just a scalar in $\mathbb{K}$ and $f(x) - g(x)$ is just a linear combination of vectors and so the expression
$\lim_{x \to a} \frac{f(x)-g(x)}{x-a} $
is meaningful
In fact, would it not be the case that the normed quotient approaches the number $0$ if and only if the unnormed quotient approaches the vector $0_E$?