There is no "limiting case," you simply add increasingly smaller amounts forever.
This was an interesting thread, but I have to say there is quite a bit of confusion involving limits and their definition and rationale, especially about the "epsilon," or what some have been calling the "infintesimal." E > 0 always. NOT 0.
Some would be better served reading a calculus textbook (limits and infinite series) than attempting to disprove the calculus. I guess the problem is one of viewpoint; whether or not a (finite) limit produces a single, unique real number or not, and whether this number cannot be used like any other real number for some strange reason.
In other words, its a debate about the validity of the definition of a (finite) limit whether or not 0.999... = 1.
Overall, I have to say that reading this thread has not convinced me that the definition of a limit is flawed.