Unit 1 · Topic 1.2 Beta

What a Limit Is and How to Write It

lim as x → a, f(x) = L says that f(x) closes in on L as x approaches a from both sides.

Practice 1: Implementing Mathematical ProcessesPractice 4: Communication and Notation

Part 1 · Hook

Why this matters

In Topic 1.1, a falling stone's average speeds 48, 40, 33.6, 32.16, 32.016 kept closing in on 32. Calculus turns "closing in" into a precise idea with its own notation, the limit. Every later idea in this course is built on it, so learning to write it and read it now pays off in every unit.

Part 2 · Before you start

What this builds on

Part 3 · Prerequisite check

Quick check before you start

1. The average speed of a stone over $[1, 1 + h]$ is $32 + 16h$ . What does it close in on as $h$ shrinks toward 0?

$32$
$48$
$16$
$0$

Show the answer

$16h$ shrinks toward 0, leaving 32.

Correct: $32$ :
$48$ :
$16$ :
$0$ :

2. On a graph, an open circle at $(2, 4)$ means

the point $(2, 4)$ is not on the graph
$f(2) = 4$
$f(4) = 2$
the graph crosses the $x$ -axis at 2

Show the answer

An open circle marks a point left out of the graph.

Correct: the point $(2, 4)$ is not on the graph:
$f(2) = 4$ :
$f(4) = 2$ :
the graph crosses the $x$ -axis at 2:

Part 4 · See it

See it first

A curve y = g(x) rises toward an open circle at (2, 4) from both sides, with arrows showing the heights heading to 4. The value g(2) is a separate filled dot at (2, 1). Panels state that the limit of g(x) as x approaches 2 is 4 and the value g(2) is 1. — The limit of $g(x)$ as $x \to 2$ is 4, where the curve heads; the value $g(2)$ is 1, the filled dot. A limit never looks at the point itself. LevlPrep original diagram.

Part 5 · Step by step

How it works, step by step

Let $x$ get close to $a$ from both sides, but never equal $a$ watch where the outputs $f(x)$ go
If the outputs close in on a single number $L$ we write $\lim_{x \to a} f(x) = L$ : the limit of $f(x)$ as $x$ approaches $a$ is $L$
The definition never uses $x = a$ itselfthe limit can differ from $f(a)$ , or exist when $f(a)$ is not defined
A limit can be read from a graph, a table or a formulathe next four topics teach each way

Part 6 · Key ideas

Key ideas

$\lim_{x \to a} f(x) = L$ means: as $x$ gets close to $a$ (from both sides, $x \ne a$ ), $f(x)$ gets close to $L$ , as close as we like.
The limit at $a$ and the value $f(a)$ are different questions. A hole can have a limit and no value; a filled dot off the curve gives a value that is not the limit.
Read the notation from the bottom up: the input heads to $a$ , the output heads to $L$ .
Write "lim" with its arrow on every line until the line where you substitute and get a number; then drop it.

Part 7 · Misconception

A common mistake

The wrong idea: $\lim_{x \to a} f(x)$ is just $f(a)$ , the value of the function at $a$ .

What actually happens: The limit describes where $f(x)$ heads for inputs near $a$ , never at $a$ . It equals $f(a)$ for many functions, but not when the graph has a hole or a dot off the curve, and it can exist when $f(a)$ does not.

Part 8 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Graph

The graph of $f$

The graph of

f

-1 \le x \le 4

follows a curve with an open circle at

(1, 1)

. The function's value at

x = 1

is the separate filled dot at

(1, 3)

1. What is $\lim_{x \to 1} f(x)$ ?

Show the answer

As $x$ gets close to 1 from either side, the heights of the graph get close to 1, the height of the open circle. The filled dot at $(1, 3)$ is the value $f(1)$ , which the limit ignores.

Correct: $1$ : The curve heads to height 1 from both sides.
$3$ : 3 is $f(1)$ , the value at $x = 1$ . A limit is about where the heights head as $x$ approaches 1.
$2$ : No part of the graph near $x = 1$ heads to 2.
$0$ : The graph near $x = 1$ stays at heights near 1, not 0.

2. What is $f(1)$ ?

$3$
$1$
$f(1)$ is not defined
$2$

Show the answer

$f(1)$ is the height of the filled dot above $x = 1$ : 3.

Correct: $3$ : The filled dot is at $(1, 3)$ .
$1$ : The open circle at $(1, 1)$ is not on the graph; 1 is the limit, not the value.
$f(1)$ is not defined: There is a filled dot at $x = 1$ , so $f(1)$ is defined.
$2$ : No point of the graph is at $(1, 2)$ .

3. What is $\lim_{x \to 3} f(x)$ ?

$3$
$1$
$5.5$
$2$

Show the answer

Near $x = 3$ the graph is an ordinary unbroken curve through $(3, 3)$ , so the heights close in on 3. Here the limit and the value $f(3) = 3$ agree.

Correct: $3$ : The curve passes through $(3, 3)$ with no gap.
$1$ : 1 is the limit at $x = 1$ , not at $x = 3$ .
$5.5$ : 5.5 is the height at $x = 4$ .
$2$ : Read the height above $x = 3$ : it is 3.

4. Which statement about $f$ is true?

$\lim_{x \to 1} f(x) \ne f(1)$
$\lim_{x \to 1} f(x)$ $= f(1)$
$\lim_{x \to 3} f(x) \ne f(3)$
$\lim_{x \to 4} f(x) = 3$

Show the answer

At $x = 1$ the limit is 1 but the value is 3, so they are not equal.

Correct: $\lim_{x \to 1} f(x) \ne f(1)$ : 1 and 3 differ.
$\lim_{x \to 1} f(x)$ $= f(1)$ : The limit is 1 and $f(1) = 3$ .
$\lim_{x \to 3} f(x) \ne f(3)$ : At $x = 3$ both the limit and the value are 3.
$\lim_{x \to 4} f(x) = 3$ : Near $x = 4$ the curve is at height about 5.5, not 3.

5. Which sentence reads $\lim_{x \to 4} f(x) = 7$ correctly?

As $x$ approaches 4, $f(x)$ approaches 7.
As $x$ approaches 7, $f(x)$ approaches 4.
$f(4) = 7$ .
As $f(x)$ approaches 4, $x$ approaches 7.

Show the answer

The number under "lim" is where the input goes ( $x \to 4$ ); the number after the equals sign is where the output goes (7).

Correct: As $x$ approaches 4, $f(x)$ approaches 7.: Input heads to 4, output heads to 7.
As $x$ approaches 7, $f(x)$ approaches 4.: The input and output numbers are swapped.
$f(4) = 7$ .: A limit statement says nothing certain about the value at $x = 4$ .
As $f(x)$ approaches 4, $x$ approaches 7.: The roles of $x$ and $f(x)$ are swapped.

6. A student says: " $\lim_{x \to 2} h(x) = 5$ , so $h(2) = 5$ ." Is the student right?

Not necessarily: $h(2)$ could be a different number or not defined at all.
Yes: a limit is the value of the function at the point.
No: if the limit is 5, then $h(2)$ is some other number.
Yes: whenever a limit exists, the function is defined there.

Show the answer

The limit is about inputs near 2, never at 2. $h(2)$ might be 5, might be another number, or might not exist. The graph of $f$ in this topic, with limit 1 and value 3 at $x = 1$ , is an example.

Correct: Not necessarily: $h(2)$ could be a different number or not defined at all.: The limit ignores $x = 2$ itself.
Yes: a limit is the value of the function at the point.: The limit and the value are different ideas; they agree only for some functions.
No: if the limit is 5, then $h(2)$ is some other number.: They can be equal; for many functions they are.
Yes: whenever a limit exists, the function is defined there.: A function with a hole has a limit at the hole but no value there.

7. A function $k$ has $k(3) = 8$ , and as $x$ approaches 3 from either side, the values $k(x)$ close in on 6. Which is correct?

$\lim_{x \to 3} k(x) = 6$ and $k(3) = 8$
$\lim_{x \to 3} k(x) = 8$ and $k(3) = 6$
$\lim_{x \to 3} k(x) = 8$ and $k(3) = 8$
$\lim_{x \to 6} k(x) = 3$ and $k(3) = 8$

Show the answer

The limit is where the values head (6); the value at 3 is what $k$ actually gives there (8).

Correct: $\lim_{x \to 3} k(x) = 6$ and $k(3) = 8$ : Limit 6 from the trend, value 8 at the point.
$\lim_{x \to 3} k(x) = 8$ and $k(3) = 6$ : The two numbers are swapped.
$\lim_{x \to 3} k(x) = 8$ and $k(3) = 8$ : The value at the point does not set the limit.
$\lim_{x \to 6} k(x) = 3$ and $k(3) = 8$ : The input target is 3, not 6.

Part 9 · Summary

Summary

\lim_{x \to a} f(x) = L

says that

f(x)

closes in on

L

x

approaches

a

from both sides. The value at

a

never enters: the limit can equal

f(a)

, differ from it, or exist where

f(a)

is undefined. Limits can be found from graphs, tables and formulas, and the rest of Unit 1 shows how.

Part 10 · Up next

What comes next

Part 11 · Connections

Why this matters

What this builds on

Quick check before you start

See it first

How it works, step by step

Key ideas

A common mistake

Check yourself

The graph of ff

Summary

What comes next

Connections

The graph of $f$