Zero and infinity are special numbers, and often used as proxies for “very small” and “very large”. Taking them literally when they’re intended as approximations can lead to difficulties in understanding. Usually, macroscopic physical properties don’t achieve those extreme values.
In physical systems, zero and infinity are often signals that the model we used to calculate them is broken. This is seen, for example, in modeling black holes with General Relativity. When building a black hole model with our best theory of gravity, one finds that a finite mass (the original star) gets compressed down to a point of zero size, with an infinitely-strong gravitational force. That point is hidden inside an “event horizon” which makes it impossible to observe, and which seemingly cuts it off from the rest of the universe. Whatever happens at the center would seemingly have no effect on anything else, ever. This causes a great deal of consternation among physicists. Other warning signs that the model has become unreliable include that the large forces and small spatial scales near that singularity are many orders of magnitude beyond the regime in which theories of gravity have been developed and tested. The spatial scales are also small enough that quantum mechanics is expected to have a large influence… and even small enough that quantum mechanics itself may have problems. Also, unlike the convergence of lines of longitude at the Earth’s poles, gravitational singularity appears intrinsic to the theory, and cannot be removed by switching to a more convenient basis. All of these are signals that General Relativity’s model of gravity can’t be applied to the center of a black hole.
The most honest approach to problems in which models fail like this is to label, as best you can, the boundaries of your model. The event horizon itself is one such boundary, although plausibly one of the weaker ones: Although we can’t directly observe what happens inside the event horizon of a black hole, it seems likely that the equations of physics are continuous there, and that an observer crossing the event horizon wouldn’t notice a sudden change. One can also estimate the scales at which quantum mechanics should take over from general relativity. (Notably, although I’m nominally talking about spatial scales here, scales of momentum and energy and acceleration and other quantities are also relevant, and we can construct scenarios which are “classical” for some scales and “quantum” for others.) Alternately, we can try to average over the pathological behavior in some way. The details of how you construct the average may make a large difference in the predictions you come up with. Alternately, you could stick with your best model and run a large number of simulations that approach the pathological regime, but don’t stray into it, and try to summarize the results.