To discuss about the constants $a_n$ and $b_n$ in the
Fisher-Tippett-Gnedenko theorem, let us assume that $X_k$ is a
sequence of i.i.d. r.vs with distribution $F(x)$; we consider $M_n:=
\max\{X_1,\,X_2,\,\dots,\,X_n\}$ so that $F_{M_n}(x) = F(x)^n$. The
constants are such that $(M_n - b_n) / a_n$ converges to a
non-degenerate distribution, say $G(x)$, or in other words: $F(a_n \,x
+ b_n)^n \to G(x)$ for all $x$. Note that the constants are not
uniquely determined: sequences $a_n'$ and $b_n'$ with $a_n' \sim a_n$
and $(b_n - b_n') / a_n \to 0$ can be used as well, with an unchanged
non-degenerate limit distribution.
As a general rule, one makes use of the the so-called tail-quantile
function $U(t)$. When $F(x)$ is continuous, $U(t)$ is
defined for $t > 1$ with its value given by
$$
1 - F(U) = 1 / t.
$$
In the vocabulary of applications, $U(t)$ is nothing but the
$t$-years return level; the scalar $U$ then has the same physical
dimension as the r.vs $X_i$ (length, time, temperature, ...).
Now $U(n)$ gives an order of magnitude of $M_n$. Indeed with
$U_n:= U(n)$
$$
F_{M_n}(U_n) = F(U_n)^n = \left\{ 1 - [1 - F(U_n)] \right\}^n
= \left\{ 1 - 1 / n \right\}^n \approx e^{-1},
$$
so clearly $U(n)$ is in the bulk of the distribution of $M_n$ for
large $n$. By either subtracting $U(n)$ or by dividing by $U(n)$ we can
hope to scale $M_n$ so that it nearly has a fixed distribution for
large $n$. But the choice depends on the type of tail i.e. of the
domain of attraction of $X$.
The two simple cases are Weibull and Fréchet. Indeed with $\omega$
being the upper end-point of $F$, it can be proved that
\begin{equation}
\text{Weibull, type III} \qquad
\frac{M_n - \omega}{U(n) - \omega} \to G(x),
\end{equation}
and
\begin{equation}
\text{Fréchet, type II} \qquad \frac{M_n - 0}{U(n)} \to G(x).
\end{equation}
The Gumbel case is more complicated and quite subtle. The value $U(n)$
is then subtracted to $M_n$, i.e. used as $b_n$ and we need the scale
$a_n$. If $X$ turns out to have a density $f(x)$ at least near
$\omega$, we can use the hazard rate $h(x)$ and the mean residual
life $e(x)$
$$
h(x) := \frac{f(x)}{1 - F(x)}, \quad e(x) :=
\frac{\int_x^\omega [1 - F(t)] \, \text{d}t}{1 - F(x)}.
$$
Both $1/h(x)$ and $e(x)$ have the same physical dimension as $x$.
Under some mild restrictions we have then
\begin{equation}
\label{eq:Gum}
\text{Gumbel, type I} \qquad \frac{M_n - U(n)}{e(U(n))} \to G(x)
\end{equation}
One condition ensuring that this convergence holds is one of Von Mises' conditions:
the derivative of $1 / h(x)$ exists and tends to $0$ for $x \to \omega$.
For many application cases, $h(x)$ is positive and monotonic for $x$
close enough to $\omega$. Assuming this, it can be shown that $F$ is
in the Gumbel domain of attraction if and only if the product
$h(x)\times e(x)$ tends to $1$ when $x \to \omega$, and then
$e(U_n)$ can be replaced by $1 / h(U_n)$.
For many classical distributions such as normal or gamma, neither
$F(x)$ nor $U(t)$ are available in closed form. Equivalent quantities
can be used, but this requires some math.
As a final remark, note that we can have a finite upper end-point
$\omega < \infty$ and yet $F$ in the Gumbel domain of attraction. An
example is provided by the reversed Fréchet distribution, for which
the determination of the constants is a good exercise.
