  The Gumbel max-domain of attraction corresponds to a null tail index which do
not distinguish the different tail weights that might exist between
distributions within this class. The Weibull-type distributions form an
important subgroup of this latter and includes the so-called \emph{Weibull-tail
coefficient}, usually denoted \theta, that specifies the tail behavior, with
larger values indicating slower tail decay. Here we shall see that the
Weibull-type distributions present a penultimate tail behavior Fr\'echet if
\theta>1 and a penultimate tail behavior Weibull whenever \theta<1.
