Limits and
continuity

Unit 1 overview

Limits and continuity form the foundation of AP Calculus AB and BC. A limit describes what a function approaches near an input, while continuity connects that approaching behavior to the function's actual value. These ideas support the later definitions of derivatives, definite integrals and, in BC, infinite series.

The 16 lessons move from instantaneous change and limit notation to graphs, tables, algebraic techniques, continuity, asymptotes and existence arguments. Work through them in order when learning the unit, or use an individual topic card to revisit a specific skill.

Weighting ranges follow the current College Board course framework. They describe the multiple-choice section, not a guaranteed number of questions or a separate fixed proportion of the free-response section.

Key concepts you will master

• Limits and notation: interpret $\underset{x\to a}{lim}f\left(x\right)=L$ as an approaching output, which can differ from the assigned function value.
• One-sided limits: distinguish $\underset{x\to a^{-}}{lim}f\left(x\right)$ from $\underset{x\to a^{+}}{lim}f\left(x\right)$ and compare them before making a two-sided conclusion.
• Algebraic limit techniques: apply valid limit laws, direct substitution, factoring, conjugates and simplification; recognize $\frac{0}{0}$ as an indeterminate form rather than an answer.
• Graphs and tables: read local behavior, compare approaches and separate numerical estimates from exact algebraic or theoretical justifications.
• Squeeze Theorem: check the required bounds and matching outer limits before concluding the middle function has that limit. Trigonometric limit arguments use radians.
• Continuity: test the function value, the limit and their equality at an interior point, then handle endpoints with limits from within the interval.
• Discontinuities: distinguish holes, jumps and unbounded behavior. A finite removable hole can be repaired by a suitable point assignment; not every discontinuity is removable.
• Asymptotes: connect signed infinite limits near a finite input to vertical lines, and finite limits at infinity to horizontal lines. Check each relevant side or end separately.
• Intermediate Value Theorem: verify continuity on a closed interval and the endpoint-value comparison before claiming a target output exists. A strictly bracketed target gives an interior solution.

Learning objectives

By the end of the unit, aim to explain the reasoning behind each result as well as calculate it.

• Evaluate or estimate limits presented algebraically, graphically, numerically or verbally.
• Select a suitable limit method and justify its algebraic steps and domain restrictions.
• Apply limit properties and the Squeeze Theorem when their hypotheses hold.
• Determine continuity at a point and on an interval, including piecewise joins and endpoints.
• Classify discontinuities and find point values or parameters that remove finite holes when possible.
• Use signed limits to justify vertical asymptotes and end limits to identify horizontal asymptotes.
• Use the Intermediate Value Theorem to justify the existence of roots or other outputs.
• Communicate using correct notation, explicit hypotheses and a conclusion appropriate to the question.
• Interpret instantaneous change, continuous models and long-term behavior with the stated units and domain.
• Prepare for AB and BC questions by practising all representations and reviewing errors across the full unit.

Why limits and continuity matter

Unit 1 supplies a toolkit used throughout calculus. Derivative definitions and later derivative applications rely on limits; integration is introduced through limits of sums, and continuity supports important forms of the Fundamental Theorem of Calculus.

Derivatives are defined through a limit

Where this finite limit exists, it defines the instantaneous rate at that input. The quotient inside the limit is an average rate over a nonzero input change. Understanding the distinction now makes differentiation easier to interpret later.

• Integration connections: continuity is a standard sufficient hypothesis in introductory FTC statements; discontinuity does not automatically mean a function has no integral.
• Problem-solving toolkit: algebraic simplification, one-sided reasoning and theorem checks return throughout the course.
• Exam preparation: limits appear directly in Unit 1 questions and also support reasoning in later topics.
• Modeling: approaching values describe instantaneous rates, continuous change and asymptotic behavior within a model's domain.
• Mathematical reasoning: distinguish a value from a nearby limit, finite behavior from unbounded growth, and evidence from a justified conclusion.
• Later units: differentiation spans Units 2–5; integration begins in Unit 6 and continues through later applications. Infinite sequences and series are BC Unit 10.

AP Calculus Unit 1 exam strategy

A reliable approach

1. Identify the approach. Read the target input, side or infinite direction before calculating.
2. Choose and justify the method. Show the relevant limit law, factorization, inequality or theorem hypotheses.
3. Use precise language. Distinguish a finite limit, a signed infinite limit and a limit that fails to exist. State exactly what a theorem guarantees.
4. Check the original domain. Cancellation does not assign a missing function value, and a denominator zero is not by itself an asymptote proof.
5. Practise across representations. Compare formulas, graphs, tables and verbal descriptions instead of relying on one format.

For one-sided questions, the two relevant notations are:

Write the reasoning requested by the problem. A memorized response or theorem name alone does not guarantee credit; explicit hypotheses and a matching conclusion make your argument assessable.

A flexible five-week study path

The original five-week progression is a suggested self-study plan. Adjust the pace to your teacher's course sequence and the skills you already know; it is not a required timetable.

1. Week 1 · Foundations · Topics 1.1–1.4

   Build intuition for instantaneous change. Learn limit notation, then read approaching values from graphs and tables. Check that you can distinguish a limit from a function value.

2. Week 2 · Algebraic techniques · Topics 1.5–1.9

   Practise limit laws, factoring, cancellation, conjugates and method selection. Add the Squeeze Theorem and connect algebraic, graphical, numerical and verbal evidence.

3. Week 3 · Continuity · Topics 1.10–1.13

   Classify discontinuities, test point and interval continuity, check one-sided endpoints and solve finite-hole or piecewise-parameter problems.

4. Week 4 · Asymptotes and existence · Topics 1.14–1.16

   Analyze signs near poles, compare both ends at infinity and justify horizontal asymptotes. Apply IVT with explicit continuity and endpoint comparisons.

5. Week 5 · Review and practice

   Complete mixed exercises from all 16 topics, attempt a Unit 1 practice test and revisit your errors. Explain why your corrected method works before repeating a similar problem.

What you will find in the topic notes

• Definitions, formulas and theorems: organized reference material with the relevant conditions and domain details.
• Worked examples: step-by-step explanations that connect a calculation to its mathematical meaning.
• Method selection: useful recognition patterns, memory aids and efficient procedures, with their limits clearly stated.
• Common mistakes: explanations of why tempting shortcuts or conclusions fail.
• Exam-focused guidance: practice communicating calculations, theorem hypotheses and conclusions.
• Practice and review: topic exercises, worked solutions and compact summaries for revisiting the material.
• Readable mathematics: typeset fractions, exponents, limits and other notation alongside explanations.
• Visual explanations: relevant teaching graphs and diagrams within the notes, clear headings and olive/cream callouts.
• Easy navigation: topic links, in-page contents and internal connections that make the notes easier to find, bookmark and revisit.

Use a topic's reference material to review a concept, then attempt its practice before reading the solution. Diagrams and tables support the reasoning; they do not replace a theorem's hypotheses or a valid limit argument.

Choose your next step

If this is your first pass through the unit, start with Topic 1.1 and progress through the foundations before moving to algebraic techniques. If you are reviewing, choose the card for the skill you need, work through its examples and then test yourself.

Build confidence by explaining each answer: what approaches what, which conditions hold and why the conclusion follows. Consistent practice across the unit prepares you for both AB and BC calculus.