We have investigated the asymptotic behavior of normal tunnel junctions at voltages where even the best Ohmic environments start to look like RC transmission lines. In the experiments, this is manifested by an exceedingly slow approach to the linear behavior above the Coulomb gap. As expected on the basis of the quantum theory taking into account interaction with the environmental modes, better fits are obtained using 1/√V than 1/V dependence for the asymptote. These results agree with the horizon picture if the frequency-dependent phase velocity is employed instead of the speed of light in order to determine the extent of the surroundings seen by the junction.