Apostle's textbook sounds interesting, not that I read math textbooks for fun (or any reason at all if I can avoid it).
We were taught (retaught) Calculus in first ("freshman") year, first semester Pure Math using Spivak's textbook which starts from a set of axioms that define the Real numbers. While this is anything but historical, I think this is better because you aren't "explaining away" stuff. At no point were we asked to simply accept something (once the axioms have been introduced and exercised and, I guess, setting aside the rigors of language and logic).
We were initially taught Calculus (in high school) the way you describe -- series and then limits -- and it royally sucked (no-one understood it and it provided no historical context).
Apostol starts similarly, with the first chapter mostly being really basic set theory (and discussion of the reals, couple of proofs about fields, etc).
Reading both volumes of Apostol is the sort of math introduction that any college student at all interested in math should have.
We were taught (retaught) Calculus in first ("freshman") year, first semester Pure Math using Spivak's textbook which starts from a set of axioms that define the Real numbers. While this is anything but historical, I think this is better because you aren't "explaining away" stuff. At no point were we asked to simply accept something (once the axioms have been introduced and exercised and, I guess, setting aside the rigors of language and logic).
We were initially taught Calculus (in high school) the way you describe -- series and then limits -- and it royally sucked (no-one understood it and it provided no historical context).