Introduction to Limits

Two thousand years ago, Zeno teased travelers with this: to reach a door, first walk half the way, then half of what remains, then half again — an endless list of errands. Can you ever arrive? Turn his puzzle into arithmetic and you meet the limit.

Endless steps, a finite sum

Add up the distances of every step:

$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$

After $n$ steps the total is $1 - \frac{1}{2^n}$ : the distance to 1 is exactly the little piece still untraveled. The pieces shrink, the sum creeps toward 1, and it never passes 1. That target value 1 — approached forever, never overshot — is the limit of the sum. The steps never stop, yet the destination is perfectly clear.

The limit: where a function is headed

Limits belong to functions too. Look at this one (at $x = 2$ the denominator is 0, so the function is not even defined there):

$f(x) = \frac{x^2 - 4}{x - 2}$

A table: closing in on 2 from both sides

At $x = 1.9$ , $f(x) = 3.9$ ; at $x = 1.99$ , $f(x) = 3.99$ ; at $x = 1.999$ , $f(x) = 3.999$ ;
At $x = 2.1$ , $f(x) = 4.1$ ; at $x = 2.01$ , $f(x) = 4.01$ ; at $x = 2.001$ , $f(x) = 4.001$ .

Both sides race toward 4, so $\lim\limits_{x \to 2} \dfrac{x^2 - 4}{x - 2} = 4$ . Why: for $x \neq 2$ , $\dfrac{x^2 - 4}{x - 2} = \dfrac{(x-2)(x+2)}{x-2} = x + 2$ , and as $x$ approaches 2, $x + 2$ approaches 4.

The key sentence: a limit describes where a function is headed — it does not require the function to arrive. $f(2)$ does not even exist, yet the limit does. Plugging in a value and taking a limit are two different operations.

One-sided limits, in one line

Approaching from the left is written $x \to 2^-$ , from the right $x \to 2^+$ ; the full limit exists only when the two one-sided limits agree. Above, both sides give 4, so the limit is 4.

The approaching lab

The limit is the heartbeat of calculus. In Introduction to Derivatives, the tangent slope is defined exactly this way: the secant slope closes in on a value as $h$ shrinks, and that limiting value is the tangent slope. Drag the slider to shrink $h$ and watch the secant hug the tangent — you are watching a limit happen:

InteractiveApproaching Lab

Point x₀0.5Step h1.5

Secant slope

2.500

Tangent slope

1.000

Drag h toward 0 and the secant hugs the tangent — that's the derivative.

Why 0.999… equals 1

Let $x = 0.999\cdots$ . Then $10x = 9.999\cdots$ , and subtracting cancels the endless tail: $10x - x = 9$ , so $x = 1$ . In the language of limits: the gap between $0.999\cdots$ and 1 is $0.000\cdots$ — a decimal whose every digit is 0. A gap of zero means they are the same number. So $0.999\cdots = 1$ is not "close enough"; it is an exact equation.

A small routine for limits

First try plugging in. If that fails (say you get $\frac{0}{0}$ ), simplify first — factor, cancel, combine fractions — to remove the bad point, then see what the simplified expression approaches.

Check yourself

Quick quiz

1. What is 1/2 + 1/4 + 1/8 + … when added forever?
2. What is lim(x→2) (x² − 4)/(x − 2)?
3. How does 0.999… (nines forever) compare with 1?