From M-W:...
In reply to:

LIMIT: 6 a : a number whose numerical difference from a mathematical function is arbitrarily small for all values of the independent variables that are sufficiently close to but not equal to given prescribed numbers or that are sufficiently large positively or negatively b : a number that for an infinite sequence of numbers is such that ultimately each of the remaining terms of the sequence differs from this number by less than any given positive amount

The above definition explains why I think that 2.9999... illustrates the concept of a limit of 3. The limit is 3, but as x approaches the limit it becomes infinitely close to 3 without quite becoming three. Sounds like 2.99999... to me.

Bean wrote:

In reply to:

Making up an entirely different function from the one being discussed that shows that the limit OF THE PARTICULAR FUNCTION is different when approaching from above or below has no relevance to this argument, which is "how do we write the number 3?", not "can we think of a function that has a limit that is undefined at 3?". Certainly f(x) = 3/(x-3) is undefined at x = 3 but that doesn't tell you a lot about the point x = 3 on the number line..

Yes but my point has to do with those two values that are not quite three. The function serves to illustrate the difference between 2.999999... and (4-0.99999...). The point at which x=3 itself has no bearing whatsoever on my argument.

Of course the big problem here is that I seem to be picking one nit and you seem to be picking another.