The limit concept was introduced by Cauchy but lacked a rigorous definition until Weierstrass came along. In the absence of such a definition, Cauchy committed some infamous errors involving nested limits, e.g. he ascribed properties to continuous functions that actually require the stronger assumption of uniform continuity. Take into account that Cauchy's treatment of calculus was by far the most rigorous for its time, so he was hardly careless in these matters. If everything was as straightforward and obvious as you imply, neither Cauchy's nor Weierstrass's work on the foundations of calculus would have served any meaningful purpose.
The more you learn, the more you come to realize that the nature of the real numbers is deeply mysterious and not to be treated lightly. That's not to say you cannot convey many useful intuitions to students without the somewhat abstruse formal machinery.
That's not to say you cannot convey many useful intuitions to students without the somewhat abstruse formal machinery.
That's more or less what I was trying to say. There are some deeply intuitive concepts for which limits are, at the very least, a very good aproximation. It may occur in some cases that the actual case and the intuition that one has about it doesn't match, but that doesn't mean that the model is arbitrary or artificial.
The more you learn, the more you come to realize that the nature of the real numbers is deeply mysterious and not to be treated lightly. That's not to say you cannot convey many useful intuitions to students without the somewhat abstruse formal machinery.