Limit

The idea of a limit grew out of two old problems. How do you find the area of a shape with a curved edge, like a circle, when area is measured in squares? And how do you draw the tangent, the line that just touches a curve at one point? Both answers come from the same trick: replace the hard thing with simpler things (straight-sided shapes, lines through two points) that get closer and closer to it, and see what they head for.

To study limits you need what you already know of algebra (factorizing), coordinate geometry (graphs) and a little trigonometry.

The word infinity comes from the Latin infinitas: in means “not” and finis means “end” or “boundary”. So something infinite has no end and no bound: a list that never stops, a process that never finishes. Infinity is written ∞.

Infinity is not a number. You can’t find it on the number line, and you can’t do ordinary sums with it. It is an idea: “without end”. When we say the real numbers run to + ∞ on the right and −∞ on the left, we mean they keep growing (or falling) past any number you name, not that they arrive somewhere called “infinity”.

6.1.1 Division by zero

There is another way to see it. Division undoes multiplication: $\frac{6}{2} = 3$ because 2 × 3 = 6. If $\frac{5}{0}$ were some number c, then 0 × c would be 5. But 0 times any number is 0, never 5. So no number works.

A number divided by 0 is undefined: $\frac{a}{0}$ has no value when a ≠ 0. What we can say is how $\frac{a}{x}$ behaves as x gets close to 0: it grows without bound, which we write $\frac{a}{x} \to \infty$. This idea, a value “becoming infinite” near a point, is what the next grade uses to study where a graph breaks.

Calculating with infinity

Since ∞ only means “grows without end”, a sum with it is a short way of saying what happens to something that grows without end. If two quantities both grow without end, so does their sum: ∞ + ∞ = ∞. If both fall without end, (−∞) + (−∞) = −∞. Multiplying keeps the sign rule for numbers: (−∞) × (−∞) = ∞, (−∞) × ∞ = −∞. And a fixed number added to something that grows without end can’t stop it growing: 5 + ∞ = ∞.

Some combinations have no single answer, and we come back to them in 6.3: ∞ − ∞, 0 × ∞ and $\frac{\infty}{\infty}$.

Questions 1 and 2

Questions 3 and 4

Question 5

When a regular polygon is drawn inside a circle and its number of sides is made larger and larger, “close to infinitely many”, it gets as close to the circle as we like. We say the limiting shape of the polygon is the circle, and the limits of its perimeter and area are the circle’s circumference and area. The polygon never becomes a circle, since a polygon with any number of sides still has straight sides and corners; but no gap you can name stays between them. That is what a limit is about: getting as close as you like.

When one quantity gets as close as we like to a number or a shape, that number or shape is its limiting value (or limit). A limit is about closeness: where something is heading, not where it is.

Limits of sequences

A sequence is a list of numbers in order, by a rule: the nth term is f(n), with n = 1, 2, 3, …. Its limit is the number its terms get as close to as we like as n grows without end, n → ∞.

As the number of terms of the sequence grows towards infinity, its terms get very close to 0. We write: as n → ∞, f(n) → 0, that is,
$$\lim_{n \to \infty} f(n) = 0$$
read “the limit of f(n) as n tends to infinity is 0”. The limit of a sequence is the real number its terms get as close to as we like, as n → ∞. No term has to equal the limit: no $\frac{1}{n}$ is ever 0.

The sum of an infinite series

The areas of the pieces, written one after another with plus signs, make an infinite series:
$$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots$$
You can’t add infinitely many numbers one by one. Instead, add the first one, then the first two, then the first three, and so on: these are the partial sums.

The partial sums make a sequence, 0.5, 0.75, 0.875, 0.9375, 0.96875, … Each falls short of 1 by $\frac{1}{2^n}$, which heads for 0, so the sequence’s limit is 1. That limit is what we mean by the sum of the infinite series:
$$\frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \cdots = 1$$

In general, for an infinite series $\sum_{n = 1}^{\infty} a_n = a_1 + a_2 + a_3 + a_4 + \cdots$, the partial sums are
S₁ = a₁
S₂ = a₁ + a₂
S₃ = a₁ + a₂ + a₃
$S_n = a_1 + a_2 + a_3 + \cdots + a_n$
and they form the sequence S₁, S₂, S₃, S₄, … The sum of an infinite series is the limit of the sequence of its partial sums.

Question 1

Question 2

Question 3

Question 4

Question 5

If n is a natural number, find the limit of each sequence.

Question 6

Find the limit of each sequence.

Question 7

Question 8

Question 9

In Figure 6.12, the outer square has an area of 16 square units. An inner square is formed by joining the midpoints of its sides, and the same is done again and again. How many inner squares can be drawn this way?

The meaning of x → a

To find the limit of a function, you must first be clear what x → a means.

x → 2 means: x takes values very close to 2, but never exactly 2, coming from both sides.

From the left, x is a little less than 2: x = 1.5, 1.9, 1.99, 1.999, 1.9999, … We write this $x \to 2^-$ (“x tends to 2 from the left”).
From the right, x is a little more than 2: x = 2.5, 2.1, 2.01, 2.001, 2.0001, … We write $x \to 2^+$ (“from the right”).

In general, x → a means that x takes values as close to the real number a as we like, from both sides, but x is never equal to a. Values close to a from the left are written $x \to a^-$, from the right $x \to a^+$. So x → a means both $x \to a^-$ and $x \to a^+$.

The indeterminate form 0/0

So $\frac{0}{0}$ can’t be given any one value. We call it an indeterminate form: an expression whose value can’t be determined uniquely. Other indeterminate forms are $\frac{\infty}{\infty}$, 0 × ∞, ∞ − ∞, $0^0$, $\infty^0$ and $1^\infty$.

Note the difference: $\frac{5}{0}$ is undefined (no number works), while $\frac{0}{0}$ is indeterminate (every number works, so none is the answer). When putting x = a into a function gives $\frac{0}{0}$, the function has no value there, but the function’s values near a may still head for one clear number, and that is what a limit finds.

Left-hand, right-hand and two-sided limits

Here, with 2 as a and 4 as l: for the limit of f(x) to be l = 4, x must approach the real number 2 very closely from both sides, and f(x) must approach the real number 4.

The limit of a function describes its behavior near a point: as x gets as close as we like to a real number a (from both sides, never equal to a), if f(x) gets as close as we like to a real number l, then l is the limit of f(x) as x tends to a. In notation,
$$\begin{aligned} &\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) = l \\ &\Rightarrow \ \lim_{x \to a} f(x) = l \end{aligned}$$
The limit exists if the left-hand and right-hand limits are equal. If they are not equal, the limit does not exist.

Why the tables keep finding the same numbers

For a function like $\frac{x^2 - 4}{x - 2}$, the tables are a way to see the limit, but there is a quicker way to be sure of it. Since x → 2 never lets x equal 2, x − 2 is never 0, so you may cancel it:
$$\frac{x^2 - 4}{x - 2} = \frac{(x - 2)(x + 2)}{x - 2} = x + 2$$
for every x ≠ 2.
As x → 2, x + 2 → 4. The same idea works whenever putting x = a in gives $\frac{0}{0}$: factorize, cancel the factor that is 0 at a, then put a into what is left. Use it to check your tables in Exercise 6.3. (When putting a in gives a number divided by 0, as $\frac{2}{x - 2}$ at 2, there is nothing to cancel: the values grow without end, and the limit doesn’t exist.)

Question 1

Questions 2 and 3

Question 4

Complete each table and find the limit.

Question 5

Question 6

Make tables to find each limit. (Three values on each side are enough to see where f(x) heads.)

Question 7

Make tables to find each limit.

Question 8

Prove each limit.

Question 9

Use each graph to answer.

Questions 10 and 11

Pasang’s 100 m, cut into halves of what is left, is the series
$$50 + 25 + 12.5 + 6.25 + \cdots$$
Its partial sums are 50, 75, 87.5, 93.75, 96.875, … m, and after n pieces he is short of the finish by exactly the last piece, $\frac{100}{2^n}$ m, which heads for 0. So the sum of all infinitely many pieces, the limit of the partial sums, is exactly 100 m: the same series as the halving square, times 100.

The times work the same way. At a steady 6.25 m/s (100 m in 16 s), the pieces take 8 s, 4 s, 2 s, 1 s, … and their sum is 8 + 4 + 2 + 1 + ⋯ = 16 s. Infinitely many pieces of time add up to a finite time, so Pasang finishes in 16 seconds, as anyone watching could tell you.

What it means: Zeno was right that the race can be cut into infinitely many pieces, and wrong that infinitely many pieces must take for ever. An infinite series can have a finite sum, and the limit is what tells you which ones do.

Grade 9’s final exam follows CDC’s Specification Grid and Model Question for Optional Mathematics, Grade 9 (2082). There Limits and continuity have 7 marks of the 75 (12 teaching hours):

In the grid’s model paper the multiple-choice question asks what the sequence with nth term $\frac{1}{n + 1}$ heads for as n tends to infinity (0). The area’s own question gives a graph with an open circle and asks for the definition of the limit of a function, a reasoned answer about whether one integer can always be divided by another, a one-sided limit read off the graph, and whether the limit exists there, with a reason. In the questions joining areas, a limit is one mark: what a length or a coefficient heads for as an angle or a number approaches some value.