  We study the effect of s-wave cross-sections in $^{4}$He+$n$ and $^{9}$Li+$n$
elastic scattering reactions by using the Jost function method (JFM). In
$^{5}$He ($^{4}$He+$n$), the s-wave pole of the $S$-matrix does not contribute
so much to the total cross-section. On the other hand, in $^{10}$Li
($^{9}$Li+$n$), the s-wave component can not be neglected due to a relatively
strong attraction for the s-waves of the core+$n$ potential. It is shown that
the $^{9}$Li-$n$ potential, which is microscopically derived by taking into
account the pairing-blocking effect for a p-wave neutron, reproduces the s-wave
pole close to the $^{9}$Li+$n$ threshold and strongly enhances the s-wave
cross-section near the threshold.
