It's true that as d approaches zero, F approaches infinity. But this does not represent any real physical systems. Remember that d is the distance between the centers of mass of the two bodies. All massive bodies have some physical size associated with them. Two billiard balls, say, even when they're touching against each other, have centers of mass separated by a few inches. Even two neutrons within the nucleus of an atom -- about as close as classical massive particles can get -- will be separated by some finite distance. In order for d to actually be zero the two masses would have to somehow be concentrated over zero space, i.e., be infinitely dense. This violates our conception of reality.
Many physical laws break down under such extreme conditions that do not correspond to observed reality.
Besides, in the case of extremely short distances, quantum mechanics (with its bizarre rules about particles being "smeared out" over space) takes over. In the case of gravity, there really is no adequate description of behavior at extremely small distances, such as the so-called Planck length (10
-35 m.) or smaller. And there is still no theory of quantum gravity, either, to resolve questions of what happens, say, when two black holes approach each other. This is a hotbed of theoretical research.
Good question! And welcome to AnswerPool.
