Unit 1 · Limits and Continuity
The Complete AP Calculus BC Guide · AP Calculus BCUnit 1 · Limits and Continuity
- The Complete AP Calculus BC Guide
- AP Calculus BC
- 10 sections
Unit 1: Limits and Continuity accounts for 5–10% of AP Calculus BC multiple-choice content. Section I has 42 multiple-choice questions in 100 minutes and contributes 50% of the score. For Section II's 6 free-response questions in 90 minutes (50%), be ready to carry the same unit skills and representations into a complete solution. College Board assigns this unit 5–10% of AP Calculus BC multiple-choice content; use that range to plan review time, not to predict a fixed number of questions or a free-response allocation.
- How AP Calculus BC assesses this 5–10% of the multiple-choice section · Section I: 42 MCQs in 100 min, 50% · Section II: 6 FRQs in 90 min, 50% · show the model with one-sided numerical table, graph with holes jumps or asymptotes, algebraic expression with a tracked domain
- Key skills Evaluate limits from multiple representations, Classify discontinuities and continuity, Verify theorem hypotheses
- How to study for Unit 1 This page turns one-sided numerical table, graph with holes jumps or asymptotes, algebraic expression with a tracked domain into one route: compare left and right approach behavior before inspecting the defined point.
- The organizing decision separate approach behavior from point value and verify every continuity or theorem condition
What AP Calculus BC Unit 1 covers
Use this map to connect each assessed skill to the relationship or representation that makes it visible.
Evaluate limits from multiple representations
one-sided numerical table; A two-sided limit exists only when both one-sided limits agreeAPCALCBC-U1-S2Classify discontinuities and continuity
graph with holes jumps or asymptotes; Continuity at a requires f(a), a finite limit, and equality between themAPCALCBC-U1-S3Verify theorem hypotheses
algebraic expression with a tracked domain; The Intermediate Value Theorem guarantees existence on a continuous closed interval, not uniquenessUnit 1: Limits and Continuity accounts for 5–10% of AP Calculus BC multiple-choice content.
Official unit name and weighting: College Board course and exam description.
Separate approach behavior from the point value
Connect the published share to the unit model
College Board assigns this unit 5–10% of AP Calculus BC multiple-choice content; use that range to plan review time, not to predict a fixed number of questions or a free-response allocation.
Unit map language to know: limit, one-sided limit, continuity, removable discontinuity, infinite limit, vertical asymptote, Intermediate Value Theorem, Squeeze Theorem, indeterminate form, and L'Hôpital's Rule; each term is used on this page and defined again in Appendix B.
A limit concerns nearby inputs; continuity adds the defined value only after the approach is known.
The decision that organizes this unit
Define the system and choose the route before calculating
separate approach behavior from point value and verify every continuity or theorem condition
compare left and right approach behavior before inspecting the defined point
Mechanism route and repair branches
Relationships to preserve
- A two-sided limit exists only when both one-sided limits agree
- Continuity at a requires f(a), a finite limit, and equality between them
- The Intermediate Value Theorem guarantees existence on a continuous closed interval, not uniqueness
Representations to read
- one-sided numerical table
- graph with holes jumps or asymptotes
- algebraic expression with a tracked domain
Branches to reject
- reading the filled point as the limit
- calling divergent behavior a finite limit
- using IVT without continuity or a closed interval
| Key concept | Why it's hard | What scores |
|---|---|---|
| Directional limits | A two-sided answer needs both sides | States both one-sided limits and compares them |
| Continuity | The point value and nearby behavior are different evidence | Checks value, limit, and equality |
| Existence theorems | A theorem name alone proves nothing | States continuity or differentiability, a numerical bridge, and the interval |
How AP Calculus BC assesses Limits and Continuity
What a complete response must make visible
Match the task to evidence that a reader can audit, then check the most likely reasoning failure before finalizing the response.
| Task | Evidence to show | Hurdle |
|---|---|---|
| Evaluate limits from multiple representations | one-sided numerical table; A two-sided limit exists only when both one-sided limits agree | reading the filled point as the limit |
| Classify discontinuities and continuity | graph with holes jumps or asymptotes; Continuity at a requires f(a), a finite limit, and equality between them | calling divergent behavior a finite limit |
| Verify theorem hypotheses | algebraic expression with a tracked domain; The Intermediate Value Theorem guarantees existence on a continuous closed interval, not uniqueness | using IVT without continuity or a closed interval |
Resolve the Limits and Continuity evidence conflict
Carry the model from prompt to check
- Step 1Read the left-hand approach, right-hand approach, and defined point value as three separate facts.
- Step 2Both one-sided limits equal 4, so the finite two-sided limit is 4.
- Step 3Compare the limiting value 4 with the defined value f(2)=7.
- Step 4State the local repair: redefining f(2) as 4 would remove this point discontinuity.
Key terms for Unit 1: Limits and Continuity
Models, uses, and boundaries
- Evaluate Two-Sided Limits
- Equal finite one-sided limits imply the two-sided limit equals L Choose this formula when the prompt asks you to evaluate two-sided limits and the function, domain, interval, or representation matches the symbols shown. A Evaluate Two-Sided Limits solution must stop if it applies this relationship before checking its domain, interval, endpoint, sign, and stated notation conditions.
- Verify Continuity at a Point
- Continuity at a means the limit equals the defined function value Choose this formula when the prompt asks you to verify continuity at a point and the function, domain, interval, or representation matches the symbols shown. A Verify Continuity at a Point solution must stop if it applies this relationship before checking its domain, interval, endpoint, sign, and stated notation conditions.
- Apply the Intermediate Value Theorem
- The Intermediate Value Theorem guarantees an input c whose output is N Choose this formula when the prompt asks you to apply the intermediate value theorem and the function, domain, interval, or representation matches the symbols shown. A Apply the Intermediate Value Theorem solution must stop if it applies this relationship before checking its domain, interval, endpoint, sign, and stated notation conditions.
- Apply the Squeeze Theorem
- The squeeze theorem traps the limit of f between two equal outer limits Choose this formula when the prompt asks you to apply the squeeze theorem and the function, domain, interval, or representation matches the symbols shown. A Apply the Squeeze Theorem solution must stop if it applies this relationship before checking its domain, interval, endpoint, sign, and stated notation conditions.
AP Calculus BC Unit 1 FAQ
How much of AP Calculus BC does Unit 1 carry?
Unit 1: Limits and Continuity accounts for 5–10% of AP Calculus BC multiple-choice content.
What is the first move on a Limits and Continuity problem?
compare left and right approach behavior before inspecting the defined point
Which relationships should I preserve?
A two-sided limit exists only when both one-sided limits agree Continuity at a requires f(a), a finite limit, and equality between them The Intermediate Value Theorem guarantees existence on a continuous closed interval, not uniqueness
Which representations should I practice?
Practice moving among one-sided numerical table, graph with holes jumps or asymptotes, algebraic expression with a tracked domain.
What error should I check before submitting an answer?
Check for reading the filled point as the limit; calling divergent behavior a finite limit; using IVT without continuity or a closed interval.
Evidence workshop
Continue from the free model into complete practice
The full unit guide continues with the chapter’s worked examples, figures, scoring tables, and answer checks.
- Prove a zero exists without claiming uniqueness
- Check the indeterminate form before differentiating
Full unit practice. Open the complete guide for the full evidence workshop and synthesis.
Related AP Calculus BC unit guides
AP Calculus BC Exam Guide & Review
The whole exam and its official unit sequence.02Differentiation: Definition and Fundamental Properties
5–10% of the multiple-choice section03Differentiation: Composite, Implicit, and Inverse Functions
5–10% of the multiple-choice section04Contextual Applications of Differentiation
5–10% of the multiple-choice section05Analytical Applications of Differentiation
10–15% of the multiple-choice section06Integration and Accumulation of Change
15–20% of the multiple-choice section07Differential Equations
5–10% of the multiple-choice section08Applications of Integration
5–10% of the multiple-choice section09Parametric Equations, Polar Coordinates, and Vector-Valued Functions
10–15% of the multiple-choice section10Infinite Sequences and Series
15–20% of the multiple-choice sectionHow to study AP Calculus BC Unit 1
Start with the organizing decision
Before solving, restate the decision in operational terms: separate approach behavior from point value and verify every continuity or theorem condition. Your first written move should be to compare left and right approach behavior before inspecting the defined point.
Practice the same idea in several representations
Rotate through one-sided numerical table, graph with holes jumps or asymptotes, algebraic expression with a tracked domain. Use each representation to practice Evaluate limits from multiple representations, Classify discontinuities and continuity, Verify theorem hypotheses, and explain what stays invariant when the surface form changes.
Turn each error into a repair check
After every attempt, audit the response for reading the filled point as the limit; calling divergent behavior a finite limit; using IVT without continuity or a closed interval. Then redo only the first step that made the reasoning diverge, keeping units, direction, and model conditions visible.
Confirm current course details in the official College Board course and exam description for the May 2027 administration.
