Consequently, 0/0 is not only undefined, but it is a so-called indeterminate form. So it is especially troubling. Moreover, when 0/0 shows up in your problem, it means you've made a mistake in the formulation and you need to reconsider exactly what the question means.
It's actually a frequent problem on the hairier edges of physics, and there's a whole set of techniques called "renormalization" to rephrase the problems in terms that don't involve undefined quantities. These techniques are very sensitive to the precise formulation of the problem, which is why some recent scientific results (like faster-than-light neutrinos or variable values of alpha) are so puzzling: they make the renormalization techniques impossible, and you end up having to throw out all of physics!