What is a limit?

A limit asks: as x gets closer and closer to some value a, what value does f(x) approach? Notice the wording — approach, not reach. The limit can exist even where the function itself is undefined.

Take (x² − 1) / (x − 1). Plug in x = 1 and you get 0/0 — undefined. But for every x near 1 the expression simplifies to x + 1, which is heading straight for 2. So the limit as x → 1 is 2, even though the function has a hole exactly at 1.