  The long-wavelength, weak-dispersion limit of the discrete nonlinear
Schr\"odinger equation with long-range dispersion is analytically considered.
This continuum approximation is carried out irrespective of the dispersion
range and hence can be assumed exact in the weak dispersion regime. For
nonlinear Schr\"odinger equations showing finite dispersion extents, the
long-range parameter is still a relevant control parameter allowing to tune the
dispersion from short-range to long-range regimes with respect to the
dispersion extent. The long-range Kac-Baker potential becomes unappropriate in
this context owing to an "edge anomaly" consisting of vanishing maximum
dispersion frequency and group velocity(and in turn soliton width) in the
"Debye" limit. An improved Kac-Baker potential is then considered which gives
rise to a non-zero maximum frequency, and allows for soliton excitations with
finite widths in the nonlinear Schr\"odinger system subjected to the long-range
but finite-extent dispersion.
