Hello guys,
We can't mix theory and reality by stating that I becomes very large while R becomes very small, and that when R equals zero V must be zero, because that's not the way it works. Either we choose to use pure theory or we choose to allow real life external quantities (like source series resistance) to enter the picture. Since back quite a few posts the OP stated that he knew that if you allow source resistance the problem becomes much easier, so he didnt want to do that. He wanted to look at the V^2/R and the I^2*R without any real life additions like that. This means he wants to stick to pure theory, so we use pure theory and dont worry about anything else.
What this means is that when we consider V^2/R that V is a theoretical voltage source, and a theoretical source (unlike a real life source) is capable of putting out an infinite current. Yes, that's an infinite current from a true theoretical voltage source. This should make the problem of V^2/R much simpler, because V is constant and only R is allowed to vary. Thus, when R goes to zero the power goes to infinity also, and it's quite obvious.
Now looking at I^2*R, it appears to be a different story, but it's not. The reason is that again we're working with pure theory and I and R are not independent, but I depends on R and V. This enables us to use the substitution V/R for I and thus we again come up with the same equation V^2/R and that's infinite power.
As a second more intuitive approach, we can consider the change in both variables. Since they are not independent, and they have a specific relationship, we can note that as R is halved, I is doubled, and this leads to a power increase of two times.
We can write this as a new equation:
P=(x*I)^2*(R/x)
and we will find the limit of P as x approaches infinity. First though, we can simplify this to:
x*I^2*R
where now I and R are constants, and take the limit as x approaches infinity. Obviously this also comes out to infinity.
So no matter how you look at it, with a theoretical voltage source and zero resistance we get an infinite current and because the voltage is constant we get an infinite power.