Stand a few metres from a wall and walk half the remaining distance. Then half of what's left. Then half again, and again, forever. You will never actually touch the wall — every step leaves a little gap — and yet nobody watching has the slightest doubt about where you are going. Your destination is completely determined by the journey, even though no single step ever arrives.

That one thought — you can know where something is heading without it ever getting there — is the idea that unlocked calculus. Here is why it was needed. All through calculus you will meet fractions that collapse to \tfrac{0}{0} at exactly the point you care about. You cannot divide zero by zero; the arithmetic simply refuses. But you can ask a sneakier question: as the inputs creep toward that forbidden point, what value is the fraction heading toward? That question always has a fair answer, and answering it is called taking a limit.

A limit asks: 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 — the wall you're walking toward, whether or not you ever stand on it.

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, exactly the collapse we were warned about. But undefined at one point doesn't mean unknowable near it. 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. Not one probe lands on x = 1 — every input is an honest, legal one — yet together they pin the destination down beyond doubt. So \lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2.

This is the standard opening move whenever a formula misbehaves at a point: retreat to the neighbourhood, take samples from both sides, and see whether they agree on a target.

Worked example: sneak up on \dfrac{x^2 - 9}{x - 3} at x = 3

Let's run the same routine ourselves, by hand, on f(x) = \dfrac{x^2 - 9}{x - 3} as x \to 3. Substituting x = 3 gives \tfrac{9 - 9}{3 - 3} = \tfrac{0}{0} — blocked. So we build the sneak-up table: three inputs creeping in from below, three from above.

From below the outputs climb 5.9,\ 5.99,\ 5.999; from above they descend 6.1,\ 6.01,\ 6.001. Both processions are marching straight at 6, and neither side ever wavers. We conclude:

The algebra explains why the table behaved so tidily. For every x \neq 3, the numerator factors and the troublemaker cancels:

Away from x = 3 the function is the line y = x + 3, and that line passes through height 6 at x = 3. The \tfrac{0}{0} was never a verdict — it was a disguise, and the limit is the tool that sees through it. That single manoeuvre, dressed in different costumes, is how every derivative in calculus will be computed.

See it on the graph

Back to our first example. 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.

Reading a limit off a graph is a physical skill, and here is the whole of it: cover the point above x = c with a fingertip, then trace the curve toward your finger — once coming from the left, once coming from the right. The height each trace is heading for as it reaches your finger is the one-sided answer; if the two traces agree, that shared height is the limit. What is hiding under the fingertip — a point, a hole, a stray dot somewhere else — never enters into it.

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.

Worked example: when the function disagrees with its own limit

A hole is one thing — the function is simply absent there. Here is a stranger case: a function that is defined at the point, but whose value sits somewhere the curve never pointed. Define

Everywhere except x = 2, g is the familiar line. But at x = 2 exactly, someone has plucked the point off the line and relocated it up to height 5. Now sneak up on x = 2: g(1.9) = 2.9, g(1.99) = 2.99, g(1.999) = 2.999 from the left, and g(2.1) = 3.1, g(2.01) = 3.01, g(2.001) = 3.001 from the right. Every single probe uses the x + 1 rule — because every probe has x \neq 2 — and both sides squeeze onto 3. So:

The limit and the function's value are two genuinely different questions, and here they give two different answers. The limit interrogates the neighbourhood — the journey in — and the journey never once consults g(2). The lone dot at height 5 is invisible to it. (Functions where the two answers do agree at every point are the well-behaved ones — you'll meet them soon under the name continuous.)

Watch a point slide in

Here is a friendlier curve, f(x) = x + 1 near x = 2 — no hole, no relocated point, nothing suspicious. Press play and watch a point walk along the curve toward x = 2 from both directions. Its height closes in on L = 3: the left walker rises toward it, the right walker descends toward it, and the two-sided handshake meets at the same height.

For a well-behaved curve like this the limit agrees with the value — \lim_{x \to 2} f(x) = 3 = f(2) — which is exactly why such functions feel unremarkable. The limit concept earns its keep at the awkward points; here it simply confirms what your eye already believed.

Watch Sal explain it

Sal Khan walks through the same idea — a function with a gap, and a table of values sneaking up on it from both sides — at an easy pace. A second telling in a second voice is a fine way to let the idea settle.

See it explained