does 0 distance between plates mean infinite capacitance?

Question

capacitance is directly proportional to area and inversely to distance, so what is capacitance when the plates are brought together?

Answer 1

According to the formula you are scratching your head over, yes. However, it is an approximation that breaks down if the separation becomes smaller than a few atoms thick. *Every* equation in physics is an approximation to real world behavior, at best. One of the most important parts of learning physics is to understand the regime in which various equations are valid. One way to recognize when an equation is meaningless is when you get an infinity. The measurable world has no infinities. Also remember the electric field for a given voltage diverges as the plates approach, so the hold-off voltage (before breakdown) approaches zero. Basically, you just short-out the capacitor if the plates are too close.

Answer 2

There would be no capacitance since there would be no volume for it.