A limit asks a simple question: as the input x creeps closer and closer to some number c, what value does the output f(x) head toward?

Notice the word toward. We don't care what happens exactly at c — only where the function is heading as we approach. We write this as:

Read it aloud: "the limit of f(x), as x approaches c, equals L." Here L is the value the outputs close in on.

Sneak up on it with a table

Take f(x) = \frac{x^2 - 1}{x - 1}. If you try x = 1 directly you get \tfrac{0}{0} — undefined. So instead let's sneak up on x = 1 from both sides and watch the outputs. Slide the input toward 1 and read the value off.

From the left (0.9, 0.99, 0.999) and from the right (1.1, 1.01, 1.001), the outputs both squeeze toward the same number: 2. So \lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2.

See it on the graph

Why 2? Because for every x \ne 1, the fraction simplifies: \frac{x^2 - 1}{x - 1} = \frac{(x-1)(x+1)}{x - 1} = x + 1. So the graph is just the line y = x + 1 — but with a single point punched out at x = 1, drawn as a hollow circle. The function has no value at x = 1, yet the graph clearly aims straight at the height y = 2 from both sides.

This is the heart of the idea: the limit can exist even where the function itself does not. The hole doesn't matter — what matters is where the curve is pointing as you approach.

Watch a point slide in

Here is a friendlier curve, f(x) = x + 1 near x = 2. Press play and watch a point walk along the curve toward x = 2 from both directions. Its height closes in on L = 3.

Watch Sal explain it