Biting into an apple and finding a maggot is unpleasant enough, but finding half a maggot is worse. Discovering one-third of a maggot would be more distressing still: The less you find, the more you might have eaten. Extrapolating to the limit, an encounter with no maggot at all should be the ultimate bad-apple experience. This remorseless logic fails, however, because the limit is singular: A very small maggot fraction (f ≪ 1) is qualitatively different from no maggot (f = 0). Limits in physics can be singular too—indeed they usually are—reflecting deep aspects of our scientific description of the world.

In physics, limits abound and are fundamental in the passage between descriptions of nature at different levels. The classical world is the limit of the quantum world when Planck’s constant h is inappreciable; geometrical optics is the limit of wave optics when the wavelength λ is insignificant;...

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