Recent numerical simulations reveal remarkably different behavior in rotating stably stratified fluids at low Froude numbers for finite Rossby numbers as compared with the behavior at both low Froude and Rossby numbers. Here the reduced low Froude number limiting dynamics in both of these situations is developed with complete mathematical rigor by applying the theory for fast wave averaging for geophysical flows developed recently by the authors. The reduced dynamical equations include all resonant triad interactions for the slow (vortical) modes, the effect of the slow (vortical) modes on the fast (inertial gravity) modes, and also the general resonant triad interactions among the fast (internal gravity) waves. The nature of the reduced dynamics in these two situations is compared and contrasted here. For example, the reduced slow dynamics for the vortical modes in the low Froude number limit at finite Rossby numbers includes vertically sheared horizontal motion while the reduced slow dynamics in the low Froude number and low Rossby number limit yields the familiar quasigeostrophic equations where such vertically sheared motion is completely absent-in fact, such vertically sheared motions participate only in the fast dynamics in this quasigeostrophic limit. The use of Ertel's theorem on conservation of potential vorticity is utilized, for example, in studying the limiting behavior of the rotating Boussinesq equations with general slanted rotation and unbalanced initial data. Other interesting physical effects such as those of varying Prandtl number on the limiting dynamics are also developed and compared here.