We study in detail the sensitivity of the Antarctic Impulsive Transient Antenna (ANITA) to possible Formula Presented point source fluxes detected via Formula Presented-lepton-induced air showers. This investigation is framed around the observation of four upward-going extensive air shower events very close to the horizon seen in ANITA-IV. We find that these four upgoing events are not observationally inconsistent with Formula Presented-induced EASs from Earth-skimming Formula Presented both in their spectral properties as well as in their observed locations on the sky. These four events as well as the overall diffuse and point source exposure to Earth-skimming Formula Presented are also compared against published ultrahigh-energy neutrino limits from the Pierre Auger Observatory. While none of these four events occurred at sky locations simultaneously visible by Auger, the implied fluence necessary for ANITA to observe these events is in strong tension with limits set by Auger across a wide range of energies and is additionally in tension with ANITA’s Askaryan in-ice neutrino channel above Formula Presented. We conclude by discussing some of the technical challenges with simulating and analyzing these near horizon events and the potential for future observatories to observe similar events.