Infinities should not turn up as real physical quantities. A death-knell for 19th-century classical physics sounded when a once perfectly good thermodynamics model was shown to predict that hot objects should radiate an infinite spectrum of energy. This was an obviously wrong prediction. People checked the work over and over. The math was fine. The model had been working up to then. For all that, it was wrong, as Max Planck finally showed in 1900.
In this century, Richard Feynman's Nobel was for "renormalizing" the equations of Quantum Electrodynamics (QED) so that they did not generate infinities. I don't remember much more than that about it, but my point is that infinity is more of a math concept than a physics one. If you start getting infinities in physics, you start wondering why you haven't been noticing them in the world around you.