Speak Maths · Last school years

Getting closer: limits

A limit says where something is heading, even if it never arrives. Seven signs let you say it exactly.

Draft: written by an AI helper and not yet checked by a teacher.

The alphabet

Read each sign aloud, the way it is written under it.

$a_{n}$ Say: a sub nMeans: the n-th term of a list that goes on for ever
$a_{n} \to L$ Say: a sub n tends to LMeans: as n grows, the terms come closer and closer to L
$lim_{n \to \infty} a_{n} = L$ Say: the limit, as n tends to infinity, of a sub n, equals LMeans: the number the list heads for; it need not ever reach it
$\infty$ Say: infinityMeans: not a number: a sign for “without end”, as in n growing for ever
$ε$ Say: epsilonMeans: a tiny distance you are free to choose, as small as you like
$δ$ Say: deltaMeans: how close to a the input x must stay
$lim_{x \to a} f (x) = L$ Say: the limit, as x tends to a, of f of x, equals LMeans: f(x) heads for L as x nears a, whatever happens at a itself

The story: Halfway to the wall

Subtitles:

Tap a line to see its words.

$a_{1} = \frac{1}{2}$ The wall is 1 metre away. Each step covers half of what is left. After one step you have walked half a metre.
$a_{2} = \frac{3}{4}$ After two steps: three quarters of a metre.
$a_{n} = 1 - \frac{1}{2 ^{n}}$ After n steps, the gap left is one half, halved n times. So you have walked 1 minus that gap.
$a_{n} < 1$ Whatever n is, you are still short of the wall.
$a_{n} \to 1$ Yet the more steps you take, the closer a sub n comes to 1.
$n \to \infty lim a_{n} = 1$ Said as a limit: as n tends to infinity, a sub n heads for 1. Infinity only means “without end”.
$ε = \frac{1}{1000}$ Pick a tiny gap: epsilon is one thousandth of a metre.
$n \geq 10 \Rightarrow ∣ a_{n} - 1 ∣ < ε$ From step 10 on, you are always within epsilon of the wall. At step 10 the gap is 1 over 1024.
$x \to 1 lim (1 - x) = 0$ Now stand x metres from the start and walk towards the wall. As x tends to 1, the gap 1 minus x tends to 0.
$δ = ε ∣ x - 1 ∣ < δ \Rightarrow ∣ 1 - x ∣ < ε$ To keep the gap below epsilon, you only need to stay within delta of the wall, and delta equal to epsilon works.

In words this story is 823 letters long; in symbols, 88.

Read it

What does it say?

$a_{n} \to 0$

$x \to 2 lim f (x) = 5$

$n \to \infty lim \frac{1}{n} = 0$

$∣ a_{n} - L ∣ < ε$

$n \to \infty$

$x \to 3 lim (2 x + 1) = 7$

$∣ x - a ∣ < δ$

$\forall ε > 0 \exists N \forall n \geq N ∣ a_{n} - L ∣ < ε$

$a_{n} \to \infty$

$x \to a lim f (x) = f (a)$

$∣ f (x) - L ∣ < ε$

$n \to \infty lim \frac{2 n}{n + 1} = 2$

Write it

Which notation says this?

The terms a sub n come closer and closer to 5.

As n tends to infinity, one over n tends to 0.

The distance from x to 3 is less than delta.

The limit of f of x, as x tends to 1, is 3.

The terms a sub n tend to infinity.

The distance between a sub n and 2 is less than epsilon.

As n tends to infinity, 1 over 2 to the n tends to 0.

For every epsilon greater than 0 there is a delta greater than 0.

The n-th term is n over n plus 1.

As x tends to 0, x squared tends to 0.

If x is within delta of 2, then x squared is within epsilon of 4.

The terms a sub n approach L.

Build it

Tap the tiles in order. One tile does not belong.

The limit of a sub n, as n tends to infinity, equals L.

If x is within delta of a, then f of x is within epsilon of L.

The limit of 1 over n, as n tends to infinity, is 0.

For every epsilon greater than 0 there exists a delta greater than 0.

a sub n equals 1 over n, which tends to 0.

If n is at least N, then a sub n is within epsilon of L.

Challenges

Creative puzzles that use every sign of this chapter. Take your time; there is no clock.

1 How close is close?

Five terms of a_n = 1 − 1/2ⁿ, four tolerances ε. Judge each sentence: true or false?

	$ε_{1} = \frac{1}{2}$	$ε_{2} = \frac{1}{5}$	$ε_{3} = \frac{1}{10}$	$ε_{4} = \frac{1}{20}$
$a_{1}$
$a_{2}$	✓
$a_{3}$	✓	✓
$a_{4}$	✓	✓	✓
$a_{5}$	✓	✓	✓	✓

A tick means |a_n − 1| < ε: the term is within ε of 1.

$∣ a_{3} - 1 ∣ < ε_{2}$
$∣ a_{2} - 1 ∣ < ε_{3}$
$∣ a_{5} - 1 ∣ < ε_{4}$
$∣ a_{1} - 1 ∣ < ε_{1}$
$∣ a_{4} - 1 ∣ < ε_{2} \land ∣ a_{4} - 1 ∣ < ε_{3}$
$∣ a_{3} - 1 ∣ < ε_{4}$

2 Limit talk

Pair each line with what it says.

Tap a line, then its meaning.

3 The wrong direction

This is the exact meaning of a limit of a list: from some term on, all terms are within any ε of L. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

4 Who stays close to whom?

The exact meaning of a limit at a point: for every ε there is a δ so that x within δ of a puts f(x) within ε of L. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

5 Almost one

Which number does the list n/(n + 1) head for?

$a_{n} = \frac{n}{n + 1}, n \to \infty lim a_{n} = ?$

6 The hole in the graph

At x = 2 the fraction is 0/0, but the limit only looks nearby. What is it?

$x \to 2 lim \frac{x ^{2} - 4}{x - 2} = ?$

7 As small as you like

The list 1, 1/2, 1/3, … gets as close to 0 as anyone could demand. Write it two ways: with lim, and with ε. Then compare.

Your version

8 Half the way, again

Each step walks half of the distance that is left, so after n steps the distance left is a_n = 1/2ⁿ. Say what the distance tends to. Then compare.

Your version

9 How far along must you go?

The list a_n = 1/n and the target 0. For a tolerance ε, which starting points N work? Judge each sentence: true or false?

	$ε = 1$	$ε = \frac{1}{2}$	$ε = \frac{1}{10}$
$N = 1$
$N = 3$	✓	✓
$N = 10$	✓	✓
$N = 100$	✓	✓	✓

A tick means: from term N on, every term is within ε of 0, that is 1/n < ε for all n ≥ N.

$\forall n \geq 3 : ∣ a_{n} ∣ < \frac{1}{10}$
$\forall n \geq 10 : ∣ a_{n} ∣ < \frac{1}{10}$
$\forall n \geq 100 : ∣ a_{n} ∣ < \frac{1}{10}$
$\exists N \forall n \geq N : ∣ a_{n} ∣ < \frac{1}{10}$
$\forall n \geq 1 : ∣ a_{n} ∣ < 1$
$lim_{n \to \infty} a_{n} = 0$

10 Four travellers

Four lists set out; where does each one head? Judge each sentence: true or false?

	$\to 0$	$\to 2$	$\to \infty$
$a_{n} = \frac{1}{n}$	✓
$b_{n} = (- 1)^{n}$
$c_{n} = n$			✓
$d_{n} = 2 - \frac{1}{n}$		✓

A tick means the list in that row does what the column says: tends to 0, tends to 2, or grows without end.

$lim_{n \to \infty} d_{n} = 2$
$b_{n} \to 0$
$c_{n} \to \infty$
$\exists L : lim_{n \to \infty} b_{n} = L$
$∣ d_{n} - 2 ∣ = \frac{1}{n}$
$a_{n} \to 0 \land d_{n} \to 0$

11 Where do they land?

Five limits, five numbers. Pair each limit with its value.

Tap a line, then its meaning.

12 Five signs

The chapter’s signs, on their own. Pair each with its job.

Tap a line, then its meaning.

13 Just short of one

This should say: the terms 1 − 1/n tend to 1. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

14 Who answers whom

The ε-δ rule: for every ε there is a δ such that x within δ of a puts f(x) within ε of L. One tile is wrong. Find it, then fix it.

Tap the tile that is wrong.

15 The first safe term

From which term on is 1/n strictly below one hundredth? Find the smallest N that works.

$a_{n} = \frac{1}{n}, ε = \frac{1}{100}, n \geq N \Rightarrow a_{n} < ε N_{m i n} = ?$

16 The cooling cup

A cup of tea cools towards the room’s temperature T but never quite reaches it: as time t goes on without end, its temperature θ(t) tends to T. Write it in symbols. Then compare.

Your version

Tell your own

Tell a three-line story about something that gets closer and closer to a target but never arrives, using a sub n and an arrow. Then read it aloud.

Kept on this device only.

Getting closer: limits

The alphabet

The story: Halfway to the wall

Read it

Write it

Build it

Challenges

1 How close is close?

2 Limit talk

3 The wrong direction

4 Who stays close to whom?

5 Almost one

6 The hole in the graph

7 As small as you like

8 Half the way, again

9 How far along must you go?

10 Four travellers

11 Where do they land?

12 Five signs

13 Just short of one

14 Who answers whom

15 The first safe term

16 The cooling cup

Tell your own

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