Limits & Limit Notation · Notes

1.Lesson overview

Syllabus focus

AP Calculus AB syllabus reference

1.1 Introducing Calculus: Can Change Occur at an Instant?
1.2 Defining Limits and Using Limit Notation
1.3 Estimating Limit Values from Graphs
1.4 Estimating Limit Values from Tables
1.5 Determining Limits Using Algebraic Properties of Limits
1.6 Determining Limits Using Algebraic Manipulation
1.7 Selecting Procedures for Determining Limits
1.8 Determining Limits Using the Squeeze Theorem
1.9 Connecting Multiple Representations of Limits

AP Calculus BC syllabus reference

1.1 Introducing Calculus: Can Change Occur at an Instant?
1.2 Defining Limits and Using Limit Notation
1.3 Estimating Limit Values from Graphs
1.4 Estimating Limit Values from Tables
1.5 Determining Limits Using Algebraic Properties of Limits
1.6 Determining Limits Using Algebraic Manipulation
1.7 Selecting Procedures for Determining Limits
1.8 Determining Limits Using the Squeeze Theorem
1.9 Connecting Multiple Representations of Limits

By the end of this lesson you should be able to

Express and interpret limits using correct notation and estimate limits from graphs and tables.
Determine limits using algebraic properties of limits and algebraic manipulation.
Determine limits using the squeeze theorem and select an appropriate procedure for a given limit.

A limit is about nearby values, not necessarily the value at the point. Algebraic simplification can expose the value approached at a removable discontinuity.

Lesson route

Begin with the meaning of the symbols, then use the worked examples to build the precise technique required for Limits & Limit Notation.

2.Source notes and further practice

Paul's Online Math Notes: this topic develops the lesson method. Practice Problems offer additional solved practice.

Examples and assessment questions in this course are original. Assignment Problems were not used.

3.Curriculum comparison and extension

Cambridge Mathematics emphasises efficient symbolic technique. AP Calculus AB requires switching fluently between graphical, numerical and analytical evidence; IB AA HL expects the method, its conditions and its interpretation to be made explicit.

Where this lesson transfers

Course route	Mapped focus
AP Calculus AB	1.1 Introducing Calculus: Can Change Occur at an Instant?; 1.2 Defining Limits and Using Limit Notation; 1.3 Estimating Limit Values from Graphs; 1.4 Estimating Limit Values from Tables; 1.5 Determining Limits Using Algebraic Properties of Limits; 1.6 Determining Limits Using Algebraic Manipulation; 1.7 Selecting Procedures for Determining Limits; 1.8 Determining Limits Using the Squeeze Theorem; 1.9 Connecting Multiple Representations of Limits; AP Calculus AB Units 1–8
IB DP Mathematics AA HL	IB AA HL · Differentiation; IB AA HL · Integrations; IB AA HL · Series and Differential Equations

Course route

Mapped focus

AP Calculus AB

1.1 Introducing Calculus: Can Change Occur at an Instant?; 1.2 Defining Limits and Using Limit Notation; 1.3 Estimating Limit Values from Graphs; 1.4 Estimating Limit Values from Tables; 1.5 Determining Limits Using Algebraic Properties of Limits; 1.6 Determining Limits Using Algebraic Manipulation; 1.7 Selecting Procedures for Determining Limits; 1.8 Determining Limits Using the Squeeze Theorem; 1.9 Connecting Multiple Representations of Limits; AP Calculus AB Units 1–8

IB DP Mathematics AA HL

IB AA HL · Differentiation; IB AA HL · Integrations; IB AA HL · Series and Differential Equations

Cross-curriculum transfer challenge

Let $f (x) = x^{3} - 3 x$ . Use $f' (x)$ to find and classify its stationary points.

Worked reasoning. $f' (x) = 3 x^{2} - 3 = 0$ gives $x = \pm 1$ . Since $f'' (x) = 6 x$ , $(- 1, 2)$ is a local maximum and $(1, - 2)$ is a local minimum. The classification needs the second-derivative evidence.

4.The mathematical idea

A limit is about nearby values, not necessarily the value at the point. Algebraic simplification can expose the value approached at a removable discontinuity.

What changes in this lesson

Do not apply a remembered rule by appearance alone. Identify the mathematical structure first, then select the relationship that answers the stated question.

5.A limit concerns nearby values, not necessarily the point

Three graphs near x = a: a straight line with an open circle (removable), two horizontal pieces at different heights (jump), and two branches rising either side of a vertical asymptote (infinite). — Figure 1: Only the removable break leaves a limit that exists, because the one-sided limits agree even though $f(a)$ itself is missing.

The statement $lim_{x \to a} f (x) = L$ describes what $f (x)$ approaches for inputs arbitrarily close to $a$ , not the value $f (a)$ . One-sided limits must agree for a two-sided limit to exist. Direct substitution is valid for continuous expressions at the point; an indeterminate form such as $0/0$ instead asks for algebraic simplification, rationalisation or another justified method. Do not cancel a factor without recording the point excluded from the original expression.

6.Distinguish approach to a point from end behaviour

A finite-point limit depends on nearby values, not necessarily on the function value at the point. A limit as $x \to \infty$ describes end behaviour instead. For rational functions at infinity, divide numerator and denominator by the highest relevant power, tracking the sign of $x$ where even roots or absolute values occur. Infinite limits signal unbounded behaviour, not a finite number called infinity.

New worked example

Find both end limits of $f (x) = \frac{4 x ^{2} - 1}{2 x ^{2} + 3 x}$ .

1
Divide numerator and denominator by $x^{2}$ : $f (x) = \frac{4 - x ^{- 2}}{2 + 3 x ^{- 1}}$ , for $x \neq = 0$ .
2
As $x \to \infty$ and as $x \to - \infty$ , the reciprocal terms tend to zero.
3
Both limits are 2, so $y = 2$ is a horizontal asymptote at both ends.

Check and interpret

This says nothing by itself about finite-domain discontinuities or whether the graph crosses the asymptote.

7.Key relationship and its conditions

Relationship to know

x \to a lim f (x) = L ⟺ x \to a^{-} lim f (x) = x \to a^{+} lim f (x) = L

Condition before calculation

A two-sided limit does not exist when the one-sided limits differ. Do not cancel a factor before noting that the original expression was undefined at its zero.

8.Vocabulary with mathematical roles

Read these terms precisely

Term	Meaning and role in this lesson
one-sided limit	The value approached while inputs move towards a point only from the left or right.
removable discontinuity	A missing or mismatched point value where the finite two-sided limit nevertheless exists.
condition	A two-sided limit does not exist when the one-sided limits differ. Do not cancel a factor before noting that the original expression was undefined at its zero.

9.Worked example 1

Question

Evaluate $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$ .

Working and final answer

Factor: $(x - 2) (x + 2) / (x - 2)$ , so the limit is $4$ .

Read the answer as a chain of equivalences or a justified algorithmic step. The final line answers the question, not only an intermediate calculation.

10.Practise the first technique

Try before revealing a method

Evaluate $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$ .
Prompt: write the relationship or algorithm before inserting numerical values.

Check

Factor: $(x - 2) (x + 2) / (x - 2)$ , so the limit is $4$ .

If your result differs, identify the first line where the mathematical object or condition changed.

11.Worked example 2

Question

Find $lim_{x \to 0^{-}} 1/ x$ .

Working and final answer

$- \infty$ .

Why this representation is efficient

This example is not a substitute for the first one: it asks for a different feature of the same limits & limit notation structure.

12.Choose the method from the structure

Decision points

If the question gives…	Then begin by…	Check before continuing
a formula or model	labelling all variables and required output	units, domain, sign or stated constraints
a graph, table or network	extracting only mathematically labelled information	whether a value is exact, estimated, or an algorithm label
an equation or expression	rewriting it into a form which exposes the target	equivalence, excluded values and endpoints

Apply this decision process to Find $lim_{x \to 0^{-}} 1/ x$ .

13.Worked example 3

Question

Evaluate $lim_{x \to 3} (2 x + 1)$ .

Working and final answer

Direct substitution gives $7$ .

This is a useful self-check: explain in one sentence why the stated condition permits the method used.

14.Extended worked example: reason through the method

Problem

Find $lim_{x \to 3} \frac{x ^{2} - 9}{x - 3}$ and explain why substitution first fails.

Solution with reasoning

1
Direct substitution gives $0/0$ , which is indeterminate, not an answer.
2
For $x \neq = 3$ , factor and cancel: $(x^{2} - 9) / (x - 3) = (x - 3) (x + 3) / (x - 3) = x + 3$ .
3
As $x \to 3$ , the simplified expression tends to $6$ .

Interpret and check

The original quotient is undefined at $x = 3$ , yet its limit exists. This is a removable discontinuity.

15.Representation and interpretation

In Limits & Limit Notation, a symbolic answer is only one representation. Pair it with a labelled diagram, a graph/table feature, a probability or a physical interpretation when that makes the conclusion checkable.

Interpretation prompt

For Evaluate $lim_{x \to 3} (2 x + 1)$ . identify what a positive/negative value, a boundary value, or an excluded value would mean in the mathematical setting.

16.A plausible error to diagnose

Do not do this

A learner writes a result for Evaluate $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$ . but never states the condition “A two-sided limit does not exist when the one-sided limits differ. Do not cancel a factor before noting that the original expression was undefined at its zero.”.

Repair

State the required condition first, then reproduce the valid calculation: Factor: $(x - 2) (x + 2) / (x - 2)$ , so the limit is $4$ .

Mathematical communication is part of the proof that the answer applies.

17.From procedure to reasoning

The key relationship is not a mnemonic detached from meaning. It compresses the idea that a limit is about nearby values, not necessarily the value at the point. algebraic simplification can expose the value approached at a removable discontinuity.

Explain it aloud

Use the worked examples to explain why each transformation, derivative, integral, probability product, force balance, or algorithm update preserves the original problem.

18.Deliberate practice set

Three exact-form tasks

1
Evaluate $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$ .
2
Find $lim_{x \to 0^{-}} 1/ x$ .
3
Evaluate $lim_{x \to 3} (2 x + 1)$ .

Answers and method cues

1
Factor: $(x - 2) (x + 2) / (x - 2)$ , so the limit is $4$ .
2
$- \infty$ .
3
Direct substitution gives $7$ .

19.Exam-quality working

What an examiner can follow

For a multi-step Limits & Limit Notation question: (1) state the relationship/model; (2) substitute or transform in visible lines; (3) preserve the condition; (4) give the requested exact value, approximation, interval, direction or conclusion.

Use Find $lim_{x \to 0^{-}} 1/ x$ . as a model: an unsupported numerical answer loses the decision-making that makes the method valid.

20.Harder transfer task

Transfer

Change one feature of this task: Evaluate $lim_{x \to 3} (2 x + 1)$ .
Decide whether the method, valid domain, or final interpretation changes. Justify the change with the key relationship.

Feedback criterion

A strong response names the changed condition, shows the altered mathematical step, and checks that the conclusion now answers the altered question.

21.Syllabus objectives in action

1. Express and interpret limits using correct notation and estimate limits from graphs and tables.
2. Determine limits using algebraic properties of limits and algebraic manipulation.
3. Determine limits using the squeeze theorem and select an appropriate procedure for a given limit.

Map the examples

Match examples 1–3 to the objective(s) above. If an objective is not represented by a worked example, use its wording to design an additional numerical or proof-based question.

22.Checkpoint questions

Before moving on

1
State, without looking, the condition attached to
$x \to a lim f (x) = L ⟺ x \to a^{-} lim f (x) = x \to a^{+} lim f (x) = L$ .
2
Re-solve: Evaluate $lim_{x \to 2} \frac{x ^{2} - 4}{x - 2}$ .
3
Explain why $- \infty$ . is a conclusion rather than only a calculator output.
4
Create a counterexample showing what can go wrong if the stated condition is ignored.

23.Lesson summary

Core idea

A limit is about nearby values, not necessarily the value at the point. Algebraic simplification can expose the value approached at a removable discontinuity.

Key relationship to retain

x \to a lim f (x) = L ⟺ x \to a^{-} lim f (x) = x \to a^{+} lim f (x) = L

Condition to keep visible

A two-sided limit does not exist when the one-sided limits differ. Do not cancel a factor before noting that the original expression was undefined at its zero.