Unit 1 · AB & BC

The Algebra of Limits.

Limits don't break when you do arithmetic to them. Add, multiply, divide, compose, and the limit follows along, almost always. Almost.

§ 1.5 · Limit Laws & Composition

§ 1 · Why this matters

Limits behave like numbers, until they don't.

Most of the limits you'll need to compute are not exotic. They're built out of simpler pieces: a sum here, a product there, a function plugged into another function. The good news, and the content of this chapter, is that the limit operator distributes through that structure.

If you know the parts, you know the whole. Take the limit of each ingredient, then assemble. That principle is what makes limits computable instead of mysterious, and it's what every later result, from derivatives to integrals, quietly leans on.

Throughout this sectionWe assume $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = K$ both exist (and are finite). The constant $k$ is any real number.

§ 2 · The five limit laws

The same two functions, used five different ways.

Below are the canonical functions we'll keep coming back to: f with $\lim_{x \ o 2} f(x) = 3$, and g with $\lim_{x \ o 2} g(x) = 1$. Every law just says: the limit of an expression built from f and g equals the same expression built from L and K.

The two reference functions. Both limits exist as $x \ o 2$.

Constant Multiple

$\lim\, [k \cdot f(x)] = k\,L$

Pulls scalars through the limit.

Sum / Difference

$\lim\, [f \pm g] = L \pm K$

Limits add the way values do.

Product

$\lim\, [f \cdot g] = L \cdot K$

Multiplication too, same idea.

Quotient

$\lim\, \dfrac{f}{g} = \dfrac{L}{K},\ K \neq 0$

Provided the denominator's limit isn't zero.

Power · Root

$\lim\, [f(x)]^n = L^n \quad\ ext{and}\quad \lim\, \sqrt[n]{f(x)} = \sqrt[n]{L}$

Exponents pass through; n-th roots too (with the obvious sign caveats).

Why does this work?

Think of the limit as a commitment to a value. If f has committed to L and g has committed to K, then the running product f(x) · g(x) can do nothing but commit to L · K. The arithmetic on the inputs becomes the same arithmetic on their destinations.

Product as area: a 3-by-1 rectangle. The limit of the product is the area of the rectangle whose sides are the individual limits.

§ 3 · A worked example

Substitute. Simplify. Done.

The five laws together justify direct substitution for any rational function at a point where the denominator is nonzero. Here's a typical calculation, end to end.

$$\lim_{x \ o 3} \rac{x^2 + 2x}{x + 1} = \rac{\lim_{x\ o 3}(x^2 + 2x)}{\lim_{x\ o 3}(x+1)} = \rac{9 + 6}{4} = \rac{15}{4}$$

The first equality is the Quotient Law. The numerator and denominator are polynomials, and limits of polynomials are computed by substitution, that's the Sum and Product laws applied repeatedly. The denominator's limit is 4, which is nonzero, so the Quotient Law is in play and we're allowed to write what we wrote.

Watch for 0/0If the denominator's limit is zero, the Quotient Law is silent, it neither approves nor forbids. You must factor, simplify, rationalize, or apply L'Hôpital before you can read off a value. The form 0/0 is an indeterminate form: it tells you to do more work, not that the limit fails to exist.

§ 4 · Composition, the hand-off

Two machines in series.

Composition asks something different. We have $f \circ g$: feed x into g, take the output, feed it into f. As x approaches c, what does the chained output approach?

The hand-off: x → c sends g(x) → K, which sends f(g(x)) → L.

Limit of a Composition

$$\lim_{x \to c} f(g(x)) = f\!\left(\lim_{x \to c} g(x)\right) = f(K) = L$$

provided f is continuous at K.

That clause, “f is continuous at K,” is the whole game. Without it, the second machine can betray you, even if both f and g have perfectly fine limits in isolation.

The hand-off doesn't care which side you came from.

A common worry: if x → c from the left, but g is decreasing, then g(x) approaches K from the right. Does that break the chain? It doesn't, f's left and right limits both equal L when it's continuous, so a directional flip on the inside is harmless.

Try it. Drag x, then toggle the direction of g:

x1.50

1.50

g(x)

0.75

f(g(x))

2.38

approach

x → 2⁻

As x approaches 2 from the left, g(x) approaches 1 from the left, u approaches K from the left too. Direction is preserved.

The middle limit always lands at K = 1; the final output always lands at L = 3, regardless of which side g approaches K from.

Both sides, both planes.

The diagram below makes the whole symmetry visible at once. The moss-colored stream tracks x → 2⁻; the slate stream tracks x → 2⁺. Both streams transit through the bend, both arrive at L.

Two approach directions, two streams, one shared destination.

§ 5 · When the chain breaks

Three cases, only one of which fails.

The continuity hypothesis isn't decoration; it's load-bearing. Here are three scenarios that look almost identical but behave very differently.

Case A · clean

Both f and g are continuous at the relevant point.

Limit = 3Substitution works directly. This is what every standard problem looks like.

Case B · subtle

f has a removable discontinuity at K, but g's outputs avoid K near c.

Limit = 3Inputs to f only graze K = 1, never land on it. f reads its limit value, not its actual value.

Case C · broken

Same hole in f, but g hits K infinitely often near c.

Limit DNESome inputs to f see 3, some see 4. The output bounces, no commitment, no limit.

The takeawayThe composition law works whenever f is continuous at K. If it isn't, the limit might still exist (Case B), or might fail (Case C), and which one happens depends on whether g's outputs steer around the bad point or through it.

See it for yourself.

The same three cases, now interactive. Switch between them and drag x. Watch the readout: in cases A and B the output settles, in case C it refuses to.

x2.40

2.40

g(x)

1.200

f(g(x))

3.32

limit at c=2

Both pieces are continuous. The chain transmits cleanly: lim f(g(x)) = f(K) = 3. Direct substitution works.

Continuity is the difference between “limit exists” and “limit DNE.” The geometry encodes the proof.

§ 6 · Summary

What to remember.

Limit laws

For arithmetic combinations, sum, difference, constant multiple, product, quotient (denominator nonzero), power, root, the limit of the combination equals the combination of the limits. This justifies direct substitution for any rational function at a point where the denominator doesn't vanish.

Composition

For $f \circ g$, the limit hand-off works provided f is continuous at K = $\lim g$. If f isn't continuous there, the composition's limit may still exist, but only if g's outputs avoid the discontinuity near c.

The mental model

A limit is a commitment to a value. Arithmetic on commitments produces the obvious commitment. Composition transmits commitments through machines, as long as the receiving machine doesn't have a hole at the point being delivered.