Very close, yes, but never equal. When dealing with limits, the difference (epsilon) approaches but never equals 0. If it does, the limit is undefined.

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.