Example 4

As the complex variable $z$ tends to 0 one has $e^z - 1 \sim z$. More generally, if $V$ is a normed space over a valued field $K$, and $f$ is a function defined on a neighbourhood of $x_0 \in K$, with values in $V$, and admitting a derivative $f'(x_0) \neq 0$ at the point $x_0$, then, as $x$ approaches $x_0$, we have $f(x) - f(x_0) \sim f'(x_0)(x - x_0)$ (I, p. 3, def. 1).