Continuity, Discontinuities & Asymptotes · Notes

1.Lesson overview

Syllabus focus

AP Calculus AB syllabus reference

1.10 Exploring Types of Discontinuities
1.11 Defining Continuity at a Point
1.12 Confirming Continuity over an Interval
1.13 Removing Discontinuities
1.14 Connecting Infinite Limits and Vertical Asymptotes
1.15 Connecting Limits at Infinity and Horizontal Asymptotes
1.16 Working with the Intermediate Value Theorem (IVT)

AP Calculus BC syllabus reference

1.10 Exploring Types of Discontinuities
1.11 Defining Continuity at a Point
1.12 Confirming Continuity over an Interval
1.13 Removing Discontinuities
1.14 Connecting Infinite Limits and Vertical Asymptotes
1.15 Connecting Limits at Infinity and Horizontal Asymptotes
1.16 Working with the Intermediate Value Theorem (IVT)

By the end of this lesson you should be able to

Classify discontinuities and define continuity at a point and over an interval.
Determine values that remove a discontinuity.
Connect infinite limits with vertical asymptotes and limits at infinity with horizontal asymptotes.
Apply the Intermediate Value Theorem to a continuous function on an interval.

Continuity at a point requires a defined value, an existing limit, and agreement between them. Asymptotes describe behaviour, not values the graph cannot cross.

Lesson route

Begin with the meaning of the symbols, then use the worked examples to build the precise technique required for Continuity, Discontinuities & Asymptotes.

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	1.10 Exploring Types of Discontinuities; 1.11 Defining Continuity at a Point; 1.12 Confirming Continuity over an Interval; 1.13 Removing Discontinuities; 1.14 Connecting Infinite Limits and Vertical Asymptotes; 1.15 Connecting Limits at Infinity and Horizontal Asymptotes; 1.16 Working with the Intermediate Value Theorem (IVT); 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

Continuity at a point requires a defined value, an existing limit, and agreement between them. Asymptotes describe behaviour, not values the graph cannot cross.

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.Classify the behaviour at a suspicious point

Continuity at $a$ requires $f (a)$ to exist, the two-sided limit to exist, and that limit to equal $f (a)$ . A rational expression may have a removable hole if a common factor cancels, or a vertical asymptote if the denominator tends to zero while the numerator does not. End behaviour gives horizontal or oblique asymptotes and is separate from behaviour at a finite point. Inspect the original domain before simplifying.

6.Continuity is three tests at one point

Three graphs of a function near x = a. In the first a straight line has an open circle at x = a, in the second two pieces sit at different heights, and in the third two branches rise without bound either side of a dashed asymptote at x = a. — Figure 1: Each type fails a different test: the removable one lacks a value at $a$, the jump has unequal one-sided limits, and the infinite one has no finite limit.

Continuity at $a$ requires $f (a)$ to exist, $lim_{x \to a} f (x)$ to exist, and the two values to agree. At a piecewise junction, calculate left and right limits separately. A removable discontinuity has a finite common limit but a missing or mismatched point value; a jump has unequal one-sided limits; a vertical asymptote concerns unbounded behaviour. The Intermediate Value Theorem also requires continuity across the whole closed interval, not merely at its endpoints.

New worked example

Choose $k$ so that $f (x) = {(x^{2} - 4) / (x - 2), k, x \neq = 2, x = 2$ is continuous at 2.

1
For $x \neq = 2$ , factor and cancel: $(x^{2} - 4) / (x - 2) = x + 2$ .
2
The limit as $x \to 2$ is 4, even though the original quotient is undefined at 2.
3
Set the separately defined value $f (2) = k = 4$ , making limit and value agree.

Check and interpret

The cancellation computes nearby behaviour; it does not by itself assign a value at the hole.

7.Key relationship and its conditions

Relationship to know

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

Condition before calculation

For a piecewise function, calculate both one-sided limits and the stated function value. A horizontal asymptote can be crossed at finite $x$ .

8.Vocabulary with mathematical roles

Read these terms precisely

Term	Meaning and role in this lesson
continuity at a	The condition that $f (a)$ exists and equals the two-sided limit as $x$ approaches $a$ .
vertical asymptote	A line $x = a$ near which function values grow without bound on at least one side.
condition	For a piecewise function, calculate both one-sided limits and the stated function value. A horizontal asymptote can be crossed at finite $x$ .

9.Worked example 1

Question

Choose $k$ so $f (x) = x^{2}$ for $x < 2$ and $k x + 1$ for $x \geq 2$ is continuous.

Working and final answer

Left limit $4$ ; $2 k + 1 = 4$ , so $k = 3/2$ .

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

Choose $k$ so $f (x) = x^{2}$ for $x < 2$ and $k x + 1$ for $x \geq 2$ is continuous.
Prompt: write the relationship or algorithm before inserting numerical values.

Check

Left limit $4$ ; $2 k + 1 = 4$ , so $k = 3/2$ .

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

11.Worked example 2

Question

Find the vertical asymptote of $1/ (x - 5)$ .

Working and final answer

$x = 5$ .

Why this representation is efficient

This example is not a substitute for the first one: it asks for a different feature of the same continuity, discontinuities & asymptotes 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 the vertical asymptote of $1/ (x - 5)$ .

13.Worked example 3

Question

Find the horizontal asymptote of $(3 x^{2} + 1) / (2 x^{2} - 7)$ .

Working and final answer

$y = 3/2$ .

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

Classify the discontinuity of $f (x) = (x^{2} - 4) / (x - 2)$ at $x = 2$ and make it continuous.

Solution with reasoning

1
At $x = 2$ the original denominator is zero, so $f (2)$ is undefined.
2
For $x \neq = 2$ , factor to get $f (x) = x + 2$ , hence $lim_{x \to 2} f (x) = 4$ .
3
Define $f (2) = 4$ to fill the hole and make the extended function continuous.

Interpret and check

There is no vertical asymptote at $x = 2$ because nearby values approach $4$ , not infinity.

15.Representation and interpretation

In Continuity, Discontinuities & Asymptotes, 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 Find the horizontal asymptote of $(3 x^{2} + 1) / (2 x^{2} - 7)$ . 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 Choose $k$ so $f (x) = x^{2}$ for $x < 2$ and $k x + 1$ for $x \geq 2$ is continuous. but never states the condition “For a piecewise function, calculate both one-sided limits and the stated function value. A horizontal asymptote can be crossed at finite $x$ .”.

Repair

State the required condition first, then reproduce the valid calculation: Left limit $4$ ; $2 k + 1 = 4$ , so $k = 3/2$ .

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 continuity at a point requires a defined value, an existing limit, and agreement between them. asymptotes describe behaviour, not values the graph cannot cross.

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
Choose $k$ so $f (x) = x^{2}$ for $x < 2$ and $k x + 1$ for $x \geq 2$ is continuous.
2
Find the vertical asymptote of $1/ (x - 5)$ .
3
Find the horizontal asymptote of $(3 x^{2} + 1) / (2 x^{2} - 7)$ .

Answers and method cues

1
Left limit $4$ ; $2 k + 1 = 4$ , so $k = 3/2$ .
2
$x = 5$ .
3
$y = 3/2$ .

19.Exam-quality working

What an examiner can follow

For a multi-step Continuity, Discontinuities & Asymptotes 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 the vertical asymptote of $1/ (x - 5)$ . 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: Find the horizontal asymptote of $(3 x^{2} + 1) / (2 x^{2} - 7)$ .
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. Classify discontinuities and define continuity at a point and over an interval.
2. Determine values that remove a discontinuity.
3. Connect infinite limits with vertical asymptotes and limits at infinity with horizontal asymptotes.
4. Apply the Intermediate Value Theorem to a continuous function on an interval.

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
$x \to a lim f (x) = f (a)$ .
2
Re-solve: Choose $k$ so $f (x) = x^{2}$ for $x < 2$ and $k x + 1$ for $x \geq 2$ is continuous.
3
Explain why $x = 5$ . 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

Continuity at a point requires a defined value, an existing limit, and agreement between them. Asymptotes describe behaviour, not values the graph cannot cross.

Key relationship to retain

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

Condition to keep visible

For a piecewise function, calculate both one-sided limits and the stated function value. A horizontal asymptote can be crossed at finite $x$ .