  A detailed study of total, Coulomb and nuclear breakup cross sections
dependence on the projectile ground-state binding energy $\varepsilon_b$ is
presented, by considering the $^8$Li+$^{12}$C and $^8$Li+$^{208}$Pb breakup
reactions. To this end, apart from the experimental one-neutron separation
energy of $^8$Li nucleus ($\varepsilon_b=2.03$~MeV), lower values of
$\varepsilon_b$ down to $\varepsilon_b=0.01$~MeV, are also being considered. It
is shown that all breakup processes become peripheral as $\varepsilon_b\to
0.01$ MeV, which is understood as due to the well-known large spacial extension
of ground-state wave functions associated to weakly-bound projectiles. The
Coulomb breakup cross section is found to be more strongly dependent on
$\varepsilon_b$ than the nuclear breakup cross section, such that the Coulomb
breakup becomes more significant as $\varepsilon_b$ decreases, even in a
naturally nuclear-dominated reaction. This is mainly due to the long-range
nature of the Coulomb forces, leading to a direct dependence of the Coulomb
breakup on the electromagnetic transition matrix. It is also highlighted the
fact that the nuclear absorption plays a minor role for small binding when the
breakup becomes more peripheral.
