The divergence is usually in terms of particle number and not empirically troubling, in that all measurable quantities remain finite.
[1][2] (Unlike in the case of the UV catastrophe where the energies involved diverge.)
Semiclassical electromagnetic theory, or the full quantum electrodynamic analysis, shows that an infinite number of soft photons are created.
Finite transition amplitudes are obtained only by summing over states with an infinite number of soft photons.
[1][2] The zero-energy photons become important in analyzing the Bremsstrahlung radiation in the coaccelerated frame in which the charge experiences a thermal bath due to the Unruh effect.