What a limit is

This chapter starts calculus. Its first idea: instead of asking what a function equals at a point, ask what it gets close to as x gets close to that point.

A falling stone. Suppose a stone falls s = 5t² metres in t seconds. How fast is it going at exactly t = 2? Take average speeds over shorter and shorter times after t = 2:

The averages close in on 20 m/s. That value is a limit.

The meaning. lim (x→a) f(x) = l means: f(x) gets as close to l as we like when x is close enough to a, without x ever being equal to a.

There are two ways to come close to a point on the number line:

• From the left (x a little less than a): the left-hand limit, written lim (x→a⁻) f(x).
• From the right (x a little more than a): the right-hand limit, lim (x→a⁺) f(x).
• The limit exists only if both are equal (and finite).

Think of a broken camera on a road: to know what happened at that spot, you check the cameras on either side. If both show the same thing, you trust it; if they disagree, you cannot say.

Working with a small h. To find a left-hand limit, put x = a – h; for a right-hand limit, x = a + h; then let h → 0, where h is small and positive.

• f(x) = |x – 3|/(x – 3) at 3. Left: |–h|/(–h) = h/(–h) = –1. Right: h/h = 1.
• They differ, so the limit at 3 does not exist.
• We may cancel h/h because h is close to 0 but not 0.

Limit and value are different things.

• (x² – 9)/(x – 3) is not defined at x = 3 (it gives 0/0), but for x ≠ 3 it equals x + 3, so the limit is 6.
• If f(x) = x² for x ≠ 1 but f(1) = 5, the limit at 1 is 1, while f(1) = 5.

Finding unknowns. f(x) = a + 2x for x < 1, f(1) = 5 and f(x) = b – x for x > 1. If the limit at 1 equals f(1): from the left a + 2 = 5, so a = 3; from the right b – 1 = 5, so b = 6.

Limits behave well with arithmetic: the limit of a sum, difference, product or quotient is the sum, difference, product or quotient of the limits (for a quotient, the bottom limit must not be 0), and a constant can be taken outside.