Limits of functions

A limit asks a careful question: as the input gets arbitrarily close to some point a, what value do the outputs settle towards? Crucially, it never asks what happens at a. The function may be undefined there, or defined with some unrelated value, and the limit is unaffected either way.

That deliberate blindness is the whole point. Consider (x² − 1)/(x − 1) at x = 1: substituting gives 0/0, which says nothing. But for every x other than 1 the expression cancels to x + 1, so as x closes in the outputs close in on 2. The limit is 2 even though the function has a hole there. This is exactly the situation a derivative lives in — a difference quotient is 0/0 at the point of interest — which is why limits had to be made precise before calculus could be.

Two over-corrections are worth avoiding. First, for well-behaved functions the limit does equal the value: for f(x) = 5 the limit at any point is 5, and "limits are never reached" is simply false. Functions where limit and value agree everywhere are precisely the continuous ones, which is how continuity gets defined. Second, a limit failing to exist is a real possibility — the outputs may head to different values from the left and the right, or grow without bound — and that is informative rather than a failure of technique.