limit of a function
Imagine walking toward a doorway. The limit of a function is the place your value is heading as your input gets closer and closer to a target point — even if you never quite arrive, and even if the door itself is locked (the function might be undefined or have the wrong value right at the target). The limit asks only about the approach, never about the destination itself.
Precisely, we say the limit of f(x) as x approaches a is L, written lim_{x->a} f(x) = L, when f(x) can be forced as close to L as we like by taking x close enough to a — but not equal to a. The point a need not be in the domain of f, but it must be a cluster point of the domain, so that there are inputs arbitrarily close to a to approach along.
A subtle but essential rule: the value f(a) is irrelevant to the limit. The functions g(x) = x and h(x) defined as x for x not equal to 0 and 7 at 0 both have limit 0 as x -> 0, even though h(0) = 7. This is exactly why limits, not values, are the right tool for defining continuity, derivatives, and integrals.
lim_{x->0} (sin x)/x = 1, even though (sin x)/x is undefined at x = 0. The hole at the origin does not stop the function from approaching 1.
A limit can exist where the value does not.
A function limit can be recast in terms of sequences: lim_{x->a} f(x) = L exactly when f(x_n) -> L for every sequence x_n -> a with x_n not equal to a. This is the sequential criterion, and it lets you transport everything you know about sequence limits to function limits.