Example. preliminaries [910]

If $f: (a, b) \to \mathbb{R}^{1}$ and if $f'(x_{0})$ exists for $x_{0} \in (a, b)$, i.e., the limit $$ \lim_{ x \to x_{0} } \frac{f(x) - f(x_{0})}{x - x_{0}} $$ exists, then let $h = x - x_{0}$, we have the following $$ \lim_{ h \to 0 } \frac{f(x_{0} + h) - f(x_{0}) - hf'(x_{0})}{h} = 0. $$ For a scalar-valued single variable function, the existence of the derivative is equivalent to say that $$ \frac{f(x_{0} + h) - \left( f(x_{0}) + hf'(x_{0}) \right)}{h} \to 0 $$ with respect to $h$.