and are complete sets (the completeness axiom applies). this is a property that makes it possible to evaluate limits, because we need to make things as close as possible. this is not possible with the integers, since there are gaps 1 unit in length between integers. finding limits under those conditions make no sense really, since we can't make things arbitrarily close (recall the epsilon-delta definition of the limit of a function)
But the definition of the limit (epsilon-delta) is based on ? So we cant really use a definition of a limit that is defined in in right? So basically when you take a limit in the closest you can be is 1 unit away. So cant you restrict the definition of a limit? You are still taking a limit right?
Actually I think you can. by the Dedekind cuts. So the definition that applies in will also apply to but not vice-versa.
we can take limits in as well. even if you could come up with a construction for limits in the integers, it would be essentially meaningless. since it would give rise to things like continuity at a point where there is actually a gap. it would simply make no sense, and won't be very interesting. in any case, if you considered only integers here, the answer would still be 2, so why even bother with it?