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In astrophysics, for instance, one often wants to compute the density of a star. The math runs into a hiccup when you want to compute the density at the center of the star because density = M/V, V=(4/3) pi R^3, but R=0 at the center so you divide by zero when you want to do M divided by V. That's not allowed but in the QFT version, you can get it to tell you that the density (of the energy) is infinite everywhere due to the L=0 lattice spacing which is used to construct a continuous field. In the star, the trick is to define
and then you can define the density at the center of a star as a finite quantity without dividing by zero (or something pretty much like that, I might not remember the exact thing. It may need a density analogue function.) I have similar thing I cooked up for the energy of the vacuum but that is like 1950s stuff and I'm still way before that in my survey of the development of the problems which remain unsolved today.
