Unit 1 · Limits and Continuity
Unit 1 · Limits and Continuity
- 10–15% of the multiple-choice section
- 5 original figures
- clean-room review
This guide organizes Limits and Continuity around one repeatable exam decision: decide what a function approaches before deciding whether a point value or continuity claim is relevant. In Limits and Continuity, formulas and vocabulary belong to an evidence chain rather than an isolated recall list.
- Decision: decide what a function approaches before deciding whether a point value or continuity claim is relevant.
- Representation: move deliberately among a numerical table approaching from both sides, a graph with holes, jumps, or vertical asymptotes, an algebraic expression simplified only after its domain is tracked.
- Limits and Continuity response standard: show the setup in standard notation, carry units through contextual quantities, and justify conclusions from the named theorem or representation.
What Limits and Continuity covers
The frozen taxonomy groups Limits and Continuity into 4 exam-facing skill routes. Each Limits and Continuity route keeps official topic ownership inside this unit.
Where Limits and Continuity sits on the exam
College Board assigns Limits and Continuity 10–15% of AP Calculus AB multiple-choice content. This range is not a share of the total exam score and does not imply a fixed question count or an FRQ allocation.
No formula or reference sheet is supplied; the graphing calculator is required only in the designated parts. Calculator details should always be checked against the current official policy at College Board.
The decision that organizes Limits and Continuity
Start with the claim, not the formula
In Limits and Continuity, the decisive question is whether you can decide what a function approaches before deciding whether a point value or continuity claim is relevant. The prompt may look computational, but a numerical table approaching from both sides must agree with the relationship 'Two-sided limits exist only when the left- and right-hand limits agree.' before the result is defensible. Begin by trying to compare the two one-sided approach behaviors and only then inspect f(a). That move keeps a graph with holes, jumps, or vertical asymptotes paired with its stated conditions and heads off the neighboring error of substituting the plotted point for the limit.
Build an evidence chain
The Limits and Continuity evidence chain begins with the situation 'A graph has an open circle at (2, 5), a filled point at (2, 1), and both branches approach 5 as x approaches 2.' and moves through a numerical table approaching from both sides, a graph with holes, jumps, or vertical asymptotes, or an algebraic expression simplified only after its domain is tracked. Each Limits and Continuity surface should lead to one named relationship and one conclusion whose scope is visible. On a numerical table approaching from both sides, label the measured feature and direction. When the same information is recast as a graph with holes, jumps, or vertical asymptotes, preserve the reference point, units, and controlled conditions. Use an algebraic expression simplified only after its domain is tracked as the final consistency check rather than leaving the answer as calculator output.
Three relationships worth being able to explain
Two-sided limits exist only when the left- and right-hand limits agree. For Limits and Continuity, test this statement against a numerical table approaching from both sides and explicitly name which quantity changes. When those Limits and Continuity conditions are absent, give a conditional prediction instead of a numerical claim.
Continuity at x=a requires existence of f(a), existence of the limit, and equality between them. Use this Limits and Continuity connection to reconcile a graph with holes, jumps, or vertical asymptotes with an algebraic expression simplified only after its domain is tracked. A Limits and Continuity disagreement points to a sign, denominator, reference, or model error that must be diagnosed before the response is finalized.
The Intermediate Value Theorem needs continuity on a closed interval and guarantees a value, not a unique input. This relationship marks the boundary next to 'using the Intermediate Value Theorem without checking continuity.' State the extra condition or observation that the stronger claim would require, especially when the prompt supplies only one representation.
Decision route.
Decision route. For Limits and Continuity, follow the evidence in order so a skipped representation or boundary does not create an overclaim.
Read the surface before you solve Limits and Continuity
What the representation can tell you
For Limits and Continuity, first name whether the prompt gives a numerical table approaching from both sides, a graph with holes, jumps, or vertical asymptotes, or an algebraic expression simplified only after its domain is tracked. On that Limits and Continuity surface, mark axes, labels, units, direction convention, and the relevant population, system, function, market, or chemical process. Describe one visible feature, then connect it to 'Continuity at x=a requires existence of f(a), existence of the limit, and equality between them..' Keeping that Limits and Continuity observation separate from its explanation makes the inference auditable and exposes any assumption that the picture itself does not show.
Error boundaries that preserve credit
The error boundary for Limits and Continuity starts with 'substituting the plotted point for the limit': return to a numerical table approaching from both sides and restore the label or condition the shortcut erased. If a solution starts calling an infinite limit a finite existing limit, make the intermediate quantity visible on a graph with holes, jumps, or vertical asymptotes instead of carrying the step mentally. The remaining boundary is using the Intermediate Value Theorem without checking continuity. Close a Limits and Continuity response by stating what an algebraic expression simplified only after its domain is tracked establishes and what additional evidence the stronger neighboring claim would need.
Representation lab.
Representation lab. This Limits and Continuity drawing is a clean-room schematic, not official exam data; read its axes and labels before importing a memorized rule.
Limit Notation and Estimation
Recognize and route the skill
Limit Notation and Estimation is a decision cluster inside Limits and Continuity; cues include the unit's named quantities, conditions, and representations. For Limit Notation and Estimation, state the target claim in words and route it through the unit decision: decide what a function approaches before deciding whether a point value or continuity claim is relevant. Routing Limit Notation and Estimation through that decision prevents a familiar operation from answering a neighboring question.
Operate, check, and communicate
For Limit Notation and Estimation, check a numerical table approaching from both sides, then apply this relationship only when its conditions match: Two-sided limits exist only when the left- and right-hand limits agree. Keep the Limit Notation and Estimation labels, sign, and context attached to the result. The adjacent Limit Notation and Estimation error is substituting the plotted point for the limit. To repair Limit Notation and Estimation, restore the missing condition, restart from compare the two one-sided approach behaviors and only then inspect f(a), and finish with evidence, consequence, and a bounded contextual claim.
Algebraic Strategies for Limits
Recognize and route the skill
Algebraic Strategies for Limits is a decision cluster inside Limits and Continuity; cues include the unit's named quantities, conditions, and representations. For Algebraic Strategies for Limits, state the target claim in words and route it through the unit decision: decide what a function approaches before deciding whether a point value or continuity claim is relevant. Routing Algebraic Strategies for Limits through that decision prevents a familiar operation from answering a neighboring question.
Operate, check, and communicate
For Algebraic Strategies for Limits, check a graph with holes, jumps, or vertical asymptotes, then apply this relationship only when its conditions match: Continuity at x=a requires existence of f(a), existence of the limit, and equality between them. Keep the Algebraic Strategies for Limits labels, sign, and context attached to the result. The adjacent Algebraic Strategies for Limits error is calling an infinite limit a finite existing limit. To repair Algebraic Strategies for Limits, restore the missing condition, restart from compare the two one-sided approach behaviors and only then inspect f(a), and finish with evidence, consequence, and a bounded contextual claim.
Continuity Definitions and Removable Discontinuities
Recognize and route the skill
Continuity Definitions and Removable Discontinuities is a decision cluster inside Limits and Continuity; cues include the unit's named quantities, conditions, and representations. For Continuity Definitions and Removable Discontinuities, state the target claim in words and route it through the unit decision: decide what a function approaches before deciding whether a point value or continuity claim is relevant. Routing Continuity Definitions and Removable Discontinuities through that decision prevents a familiar operation from answering a neighboring question.
Operate, check, and communicate
For Continuity Definitions and Removable Discontinuities, check an algebraic expression simplified only after its domain is tracked, then apply this relationship only when its conditions match: The Intermediate Value Theorem needs continuity on a closed interval and guarantees a value, not a unique input. Keep the Continuity Definitions and Removable Discontinuities labels, sign, and context attached to the result. The adjacent Continuity Definitions and Removable Discontinuities error is using the Intermediate Value Theorem without checking continuity. To repair Continuity Definitions and Removable Discontinuities, restore the missing condition, restart from compare the two one-sided approach behaviors and only then inspect f(a), and finish with evidence, consequence, and a bounded contextual claim.
How the AP Calculus AB assesses Limits and Continuity
Unit ranges describe the multiple-choice section only. Free-response work can combine content across units, so no per-unit FRQ share is inferred.
| Item | Weight / count | What it means |
|---|---|---|
| Multiple choice | 42 questions · 100 minutes · 50% | Part A has 29 no-calculator questions in 62 minutes; Part B has 13 calculator-required questions in 38 minutes. |
| Free response | 6 questions · 90 minutes · 50% | Two calculator-required questions precede four no-calculator questions; prompts are in Bluebook and responses are handwritten. |
| Calculator | Part-scoped graphing calculator | Calculator commands do not replace standard mathematical setup or justification. |
| Unit weight | 10–15% of the multiple-choice section | This published range applies to multiple choice, not to a promised count or an FRQ allocation. |
| Response evidence | Represent · relate · verify | Show the setup in standard notation, carry units through contextual quantities, and justify conclusions from the named theorem or representation. |
Choose the first defensible move in Limits and Continuity
Read each question as it appears on the exam paper, choose your answer, then tap it to see the feedback.
Practice 1: Separate a limit from a point value
The graph of f has an open circle at (2, 5), a filled point at (2, 1), and both branches approach y = 5 as x approaches 2.
Which statement is correct?
Choose the best answer.
A. The limit as x approaches 2 of f(x) is 1 because f(2) = 1.
❌ Not quite 😭 This substitutes the filled point for the limit. Limits depend on nearby approach behavior, not the value at the input.
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B. The limit as x approaches 2 of f(x) is 5, but f is not continuous at x = 2 because f(2) = 1.
✅ Correct! 😊 Both one-sided limits equal 5, so the two-sided limit exists; continuity fails because that limit does not equal f(2).
C. The limit does not exist because the graph has an open circle at x = 2.
❌ Not quite 😭 An open circle does not prevent a limit when both sides approach the same output. Here both approach 5.
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D. The limit is 5, so f is continuous at x = 2 regardless of f(2).
❌ Not quite 😭 Continuity also requires the function value to equal the limit. The filled point gives f(2) = 1, not 5.
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Full explanation
- Step 1Read the left-hand approach: f(x) approaches 5 as x approaches 2 from the left.
- Step 2Read the right-hand approach: f(x) also approaches 5 from the right.
- Step 3Thus the two-sided limit is 5, while the filled point gives f(2) = 1.
- Step 4Because the limit and function value differ, f is not continuous at x = 2.
Practice 2: Track the excluded input after factoring
For x ≠ 3, let g(x) = (x² − 9)/(x − 3).
Which statement correctly describes g near x = 3?
Choose the best answer.
A. The limit as x approaches 3 of g(x) is 6, but g(3) is undefined.
✅ Correct! 😊 Factoring gives g(x) = x + 3 only for x ≠ 3, so nearby values approach 6 while the original function has a hole.
B. g(3) = 6 because the factors cancel.
❌ Not quite 😭 Cancellation produces an equivalent expression only where x − 3 is nonzero. It does not define the original function at x = 3.
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C. The limit does not exist because direct substitution gives 0/0.
❌ Not quite 😭 The form 0/0 is indeterminate, not a conclusion. Factoring reveals the nearby behavior and a finite limit.
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D. The limit is 0 because the numerator approaches 0.
❌ Not quite 😭 The denominator also approaches 0, so the numerator alone does not determine the quotient's limit.
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Full explanation
- Step 1Factor x² − 9 as (x − 3)(x + 3).
- Step 2For x ≠ 3, cancel x − 3 to obtain g(x) = x + 3.
- Step 3Nearby values therefore approach 6 as x approaches 3.
- Step 4The original restriction remains, so g(3) is undefined.
Working language for Limits and Continuity
- Limit Notation and Estimation
- In Limits and Continuity, Limit Notation and Estimation names the linked decisions for recognizing the evidence, selecting a valid relationship, and stating a contextual conclusion.
- Algebraic Strategies for Limits
- In Limits and Continuity, Algebraic Strategies for Limits names the linked decisions for recognizing the evidence, selecting a valid relationship, and stating a contextual conclusion.
- Continuity Definitions and Removable Discontinuities
- In Limits and Continuity, Continuity Definitions and Removable Discontinuities names the linked decisions for recognizing the evidence, selecting a valid relationship, and stating a contextual conclusion.
- Infinite Limits, Asymptotes, and the IVT
- In Limits and Continuity, Infinite Limits, Asymptotes, and the IVT names the linked decisions for recognizing the evidence, selecting a valid relationship, and stating a contextual conclusion.
- Limits and Continuity
- The official Limits and Continuity frame that connects its frozen skill leaves through one evidence-preserving decision route for AP Calculus AB.
- evidence chain
- The Limits and Continuity sequence from observation to representation, relationship, operation, verification, and a claim limited by the available evidence.
- representation check
- A deliberate inspection of labels, axes, units, direction, population, system, or market before solving a Limits and Continuity problem.
- error boundary
- A condition that separates a warranted Limits and Continuity inference from a stronger neighboring claim that the prompt does not establish.
Limits and Continuity questions students actually ask
What is the first decision in Limits and Continuity?
Begin Limits and Continuity by deciding how to decide what a function approaches before deciding whether a point value or continuity claim is relevant. Then compare the two one-sided approach behaviors and only then inspect f(a). This keeps the Limits and Continuity target claim, given conditions, and representation aligned before arithmetic or symbolic manipulation begins.
Which representation should I draw for Limits and Continuity?
For Limits and Continuity, choose among a numerical table approaching from both sides, a graph with holes, jumps, or vertical asymptotes, an algebraic expression simplified only after its domain is tracked according to the evidence. Label the Limits and Continuity axes, units, system or population, and direction before using the drawing to justify a relationship or numerical result.
How do I repair the most common Limits and Continuity shortcut?
In Limits and Continuity, watch for substituting the plotted point for the limit. Return to the Limits and Continuity prompt, restore the skipped condition or representation, and rebuild the evidence chain from compare the two one-sided approach behaviors and only then inspect f(a) rather than patching the final line.
What makes a Limits and Continuity explanation complete?
In Limits and Continuity, a complete explanation names the governing relationship, points to the relevant evidence, states the directional or numerical consequence, and finishes in context. For Limits and Continuity, you should show the setup in standard notation, carry units through contextual quantities, and justify conclusions from the named theorem or representation.
Should I memorize every formula in Limits and Continuity?
For Limits and Continuity, memorize only what the official reference policy requires, but practice selecting and explaining every relationship. For Limits and Continuity, no formula or reference sheet is supplied; the graphing calculator is required only in the designated parts. A Limits and Continuity formula is useful only after its variables and assumptions match the prompt.
Continue through all AP Calculus AB units
A durable study loop for Limits and Continuity
Build a one-page decision map for Limits and Continuity. Put the question 'decide what a function approaches before deciding whether a point value or continuity claim is relevant?' at the center, connect it to a numerical table approaching from both sides, a graph with holes, jumps, or vertical asymptotes, an algebraic expression simplified only after its domain is tracked, and write the condition that licenses each relationship beside its arrow.
Practice Limits and Continuity representation translation in pairs. Convert a numerical table approaching from both sides into a graph with holes, jumps, or vertical asymptotes, then reverse the translation without looking. Any Limits and Continuity feature that disappears in one direction identifies a label, unit, or assumption that needs deliberate rehearsal.
Keep a Limits and Continuity error log organized by broken step instead of by problem number. When you catch substituting the plotted point for the limit, record the missing cue and the repair action. Re-solve the Limits and Continuity prompt after two days and one week using only that cue.
For timed Limits and Continuity work, spend the opening seconds framing the object and expected direction. Then solve the Limits and Continuity prompt, verify with a second representation or limiting case, and write the contextual conclusion. This Limits and Continuity routine is faster than repairing an answer built on the wrong model.
