Remark

Despite the practical usefulness of the scale $\mathscr{E}$ which we have just defined, it is easy to give examples of (H) functions which have no principal part with respect to $\mathscr{E}$. Indeed, if $f$ is an (H) function such that $f \sim a g$, where $a$ is a constant $> 0$ and $g \in \mathscr{E}$, then $\log f - \log g - \log a$ tends to 0 with $1/x$, so $\log f$ admits, relative to $\mathscr{E}$, an asymptotic expansion whose remainder tends to 0, by property c). Now, if one considers for example the (H) function $f(x) = e_2 \left( x + \frac{1}{x} \right)$ one has $\log f(x) = \exp \left( x + \frac{1}{x} \right)$, so the asymptotic expansions of $\log f$ relative to $\mathscr{E}$ are of the form

It is clear that the remainder in this expansion is equivalent to $\frac{1}{(n+1)!} \frac{e^x}{x^{n+1}}$, so does not tend to 0. Hence $f$ does not have a principal part relative to $\mathscr{E}$.