The Intuition of Limits

Look at the graph of y = (x² − 1)/(x − 1). At x = 1 the formula blows up — 0 divided by 0 — so the function has no value there; the graph has a hole. And yet, if you trace the curve closer and closer to x = 1, the y-values clearly head toward 2. The function isn't DEFINED at 1, but it has a DESTINATION there. That destination — the value f(x) approaches as x approaches a — is the LIMIT. Limits are the idea that lets calculus study motion and change without needing to actually arrive, and this hole-with-a-height is its perfect first image.