We provide the asymptotic properties of area spectral efficiency (ASE) of a downlink cellular network in the limit of very dense base station (BS) and user densities. 
This asymptotic analysis relies on three assumptions: (1) 
interference is treated as noise; (2) the BS locations are drawn from a Poisson point process; (3) the path loss function is bounded above satisfying mild regularity conditions. We consider three possible definitions of the ASE, all of which give units of bits per second per unit area.  When there is no constraint on the minimum operational SINR and instantaneous full channel state information is available at the transmitter, the ASE is proven to saturate to a constant, which we give in closed form.  For the other two ASE definitions, wherein either a minimum SINR is enforced or full CSI is not available, the ASE is instead shown to collapse to zero at high BS density. We show several familiar case studies for the class of considered path loss models, and demonstrate that our results cover most previous models and results on ultradense networks as special cases.