My professor provided the following definition for multivariate differentiation:
Definition: For a function $f: A \to \mathbb{R}^n$, $A \subseteq \mathbb{R}^m$, to be differentiable there must be a linear transformation $F$, and a function $g$, both dependent on $x$ such that $$f(x+h)-f(x)=F(h)+g(h)$$ where $$\lim_{h\to0} \dfrac{g(h)}{|h|} = 0.$$
Why do we require $\displaystyle\lim_{h\to0}\dfrac{g(h)}{|h|}=0$ rather than simply $\displaystyle\lim_{h\to0} g(h)=0$?
I've been trying to come up with an example that illustrates that $\displaystyle\lim_{h\to0}\dfrac{g(h)}{|h|}=0$ is necessary but cannot seem to think of anything. Any help would be appreciated.