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The course is built around the AP Calculus AB Unit 1 test on Limits and Continuity. A high score indicates that you are prepared for integrals, derivatives, and the entire AP exam.
15 AP-style multiple-choice questions (MCQs) covering every topic covered in Unit 1 are included in this guide, along with detailed explanations of both right and wrong answers. In order to help you grasp concepts rather than just answers, each question is in line with College Board CED subtopics and identifies typical errors.
| AP Calculus AB Unit 1 Resource | What’s Included | Practice |
| AP Calculus AB Unit 1 Practice Test (15 MCQs) | Graph-based questions with comprehensive explanations, limits, and continuity | Practice |
| Limits Practice Questions | Algebraic, graphical, and numerical evaluation of limits | Practice |
| Continuity Questions Set | Examples of continuity and discontinuity types | Practice |
| Graph-Based Limits Problems | Analyzing limits using tables and graphs | Practice |
| One-Sided Limits Practice | Limits on the left and right hands with complex situations | Practice |
| Infinite Limits & Asymptotes | Asymptotes that are vertical and behavior close to infinity | Practice |
| Common Mistakes Practice Set | The most common mistakes made by students when providing explanations | Practice |
| Mixed Unit 1 Review Test | Complete review exam covering every concept from Unit 1 | Practice |
The AP Calculus AB Unit 1 test covers Limits and Continuity — sub-topics 1.1 through 1.16 from the College Board AP Calculus AB Course and Exam Description (CED). Unit 1 accounts for 10–12% of the total AP Calculus AB exam score. Key topics tested include evaluating limits algebraically and from graphs, one-sided limits, the Squeeze Theorem, three types of discontinuity, continuity conditions, and the Intermediate Value Theorem (IVT).
Students also search for:
| AP Calculus AB Unit 1 test 15 questions | AP Calc AB limits and continuity quiz |
| AP Calculus AB Unit 1 practice test with answers | AP Calc AB Unit 1 progress check MCQ |
The mathematical concept of a limit and the formal definition of continuity are presented in AP Calculus AB Unit 1, Limits and Continuity. It serves as the entry point to all of calculus. Students struggle with derivatives (Unit 2) and integrals (Unit 6) if they don’t grasp Unit 1..
Limit Definition :The value that a function f(x) approaches as the input x approaches a particular value c is described by a limit. expressed as follows: lim(x→c) f(x) = L, where L is the value that is approached rather than the value at c.
Continuity Definition : If the following three conditions are met, a function f is continuous at x = c: (1) f(c) is defined; (2) lim(x→c) f(x) exists; and (3) lim(x→c) f(x) = f(c). Not just one or two, but all three must be true.
Unit 1 is broken down into 16 subtopics (1.1–1.16) in the College Board AP Calculus AB Course and Exam Description. This test consists of fifteen questions, each of which corresponds to one of these subtopics. To determine precisely which subtopics you need to review after the test, use this table.
| Sub-Topic | Title |
| 1.1 | 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 |
| 1.1 | Exploring Types of Discontinuities |
| 1.11 | Defining Continuity at a Point |
| 1.12 | Confirming Continuity Over an Interval |
| 1.13 | Removing Discontinuities |
| 1.14 | Infinite Limits and Vertical Asymptotes |
| 1.15 | Limits at Infinity and Horizontal Asymptotes |
| 1.16 | Intermediate Value Theorem (IVT) |
The AP Calculus AB exam does not include a formula sheet. Before taking the 15-question test, commit all of the formulas and theorems in the table below to memory.
| Concept | Formula / Rule | CED Sub-Topic |
| Standard Trig Limit 1 | lim(x→0) sin(x)/x = 1 | 1.5, 1.6 |
| Standard Trig Limit 2 | lim(x→0) (1 – cos x)/x = 0 | 1.5, 1.6 |
| Squeeze Theorem | If g(x) ≤ f(x) ≤ h(x) and lim g = lim h = L, then lim f = L | 1.8 |
| 3 Types of Discontinuity | Removable (hole), Jump, Infinite (vertical asymptote) | 1.10 |
| Continuity at a Point (3 Conditions) | f(c) defined AND lim exists AND lim = f(c) | 1.11 |
| Removable Discontinuity | Limit exists but f(c) is undefined or f(c) ≠ lim | 1.10, 1.13 |
| Vertical Asymptote | lim(x→c) = ±∞ when denominator → 0 and numerator ≠ 0 | 1.14 |
| Horizontal Asymptote (equal degrees) | lim(x→∞) = ratio of leading coefficients | 1.15 |
| Horizontal Asymptote (num degree < denom) | lim(x→∞) = 0 | 1.15 |
| IVT | If f continuous on [a,b] and N is between f(a) and f(b), then f(c) = N for some c in (a,b) | 1.16 |
| Category | Details |
| AP Exam Weight | 10–12% of the total score on the AP Calculus AB exam |
| Estimated MCQ Questions on Full AP Exam | Five or six questions from Unit 1 |
| FRQ Appearance | FRQ sections include IVT application and continuity justification. |
| Hardest Sub-Topics | 1.7 (Procedure Selection), 1.8 (Squeeze Theorem), 1.16 (IVT), |
| Easiest Sub-Topics | 1.3 (Graph Limits), and 1.4 (Table Limits) |
| Study Time — Strong Algebra Background | Ten to fifteen hours over a period of one to two weeks |
| Study Time — Weaker Algebra Background | 25–40 hours spread over three to four weeks |
| Recommended Daily Practice | Every day, 15–20 Unit 1 questions with thorough explanations are reviewed. |
Test Instructions: You are not allowed to use a calculator. For realistic timed practice, allot two minutes for each question, for a total of thirty minutes. Before consulting the answer key, complete all 15 questions. There is only one right response to each question.
Difficulty Key: Simple is the straightforward application of a single idea. Medium = necessitates combining two ideas or choosing the appropriate approach. Hard = necessitates applying theorems, multi-step reasoning, or spotting subtle traps.
Which of the following correctly states that the limit of f(x) as x approaches 3 is equal to 7?
A) f(3) = 7
B) lim(x→3) f(x) = 7
C) lim(x→7) f(x) = 3
D) f(7) = 3
Correct Answer: B) lim(x→3) f(x) = 7
Why This Is Correct: Option B correctly uses limit notation: lim(x→c) f(x) = L means ‘the limit of f(x) as x approaches c equals L.’ Here c = 3 and L = 7.
Why A Is Wrong: A describes the function value at x = 3, not a limit. A limit and a function value are two different things — f(3) could be undefined even if the limit exists.
Why B Is Wrong: B is correct.
Why C Is Wrong: C swaps the roles of the input value and the limit value. lim(x→7) f(x) = 3 would mean x approaches 7 and the function approaches 3 — the opposite of what is stated.
Why D Is Wrong: D states f(7) = 3, which is a function value at x = 7. This has no relation to the limit described in the question.
The graph of f shows: as x→5 from the left, f(x)→8. As x→5 from the right, f(x)→8. The point (5, 2) is on the graph. What is lim(x→5) f(x)?
A) 2
B) 8
C) Does not exist
D) 5
Correct Answer: B) 8
Why This Is Correct: A two-sided limit equals L when both the left-hand limit and the right-hand limit equal L. Both one-sided limits equal 8, so lim(x→5) f(x) = 8. The function value f(5) = 2 does not affect the limit.
Why A Is Wrong: A is the function value f(5) = 2, not the limit. Limits describe approach behavior — they are independent of the actual value at the point.
Why B Is Wrong: B is correct.
Why C Is Wrong: C (Does not exist) is wrong because both one-sided limits are equal. A limit does not exist only when the left and right limits are different, or when the function grows without bound.
Why D Is Wrong: D is the x-value being approached, not the limit value. The limit is the y-value the function approaches, not the x-value.
A table of values shows f(x) approaching 2 from both sides of x = 4. From the left: f(3.9) = 5.7, f(3.99) = 5.97, f(3.999) = 5.997. From the right: f(4.1) = 6.3, f(4.01) = 6.03, f(4.001) = 6.003. What is lim(x→4) f(x)?
A) 5.997
B) 6.003
C) 6
D) Does not exist
Correct Answer: C) 6
Why This Is Correct: Both the left-side values (5.7, 5.97, 5.997) and the right-side values (6.3, 6.03, 6.003) are converging toward 6 as x gets closer to 4. The pattern clearly shows the limit is 6.
Why A Is Wrong: A (5.997) is the closest listed left-side value to x = 4, not the limit. Students should look at the pattern of where values are heading, not just the last listed value.
Why B Is Wrong: B (6.003) is the closest listed right-side value, not the limit. Like option A, this confuses the closest table entry with the actual limit value.
Why C Is Wrong: C is correct.
Why D Is Wrong: D is wrong because both one-sided limits approach the same value (6). A limit does not exist only when left and right limits differ.
Find: lim(x→4) of (x^2 – 3x + 1).
A) 5
B) 1
C) 13
D) Does not exist
Correct Answer: A) 5
Why This Is Correct: f(x) = x^2 – 3x + 1 is a polynomial, so it is continuous everywhere. Use direct substitution: f(4) = (4)^2 – 3(4) + 1 = 16 – 12 + 1 = 5.
Why A Is Wrong: A is correct.
Why B Is Wrong: B results from computing 1 – 3 + 1 incorrectly or substituting x = 1. Check that the correct value (x = 4) is substituted.
Why C Is Wrong: C results from computing 16 – 3 = 13 and forgetting to add 1 after subtracting 3x = 12. Always complete the full arithmetic.
Why D Is Wrong: D is wrong because polynomial functions are continuous everywhere. Direct substitution always works for polynomials — the limit always exists and equals f(c).
Find: lim(x→3) of (x^2 – 9) / (x – 3).
A) 0
B) Does not exist
C) 3
D) 6
Correct Answer: D) 6
Why This Is Correct: Direct substitution gives 0/0 — an indeterminate form. Factor the numerator: x^2 – 9 = (x+3)(x-3). Cancel the (x-3) factor. The limit becomes lim(x→3) of (x+3) = 3 + 3 = 6.
Why A Is Wrong: A results from incorrectly substituting and computing 9 – 9 = 0 in the numerator without recognizing the indeterminate form. Getting 0/0 is not the answer — it is a signal to factor.
Why B Is Wrong: B is wrong. Getting 0/0 from direct substitution does not mean the limit does not exist. It means you must use algebraic manipulation (factoring, rationalizing, or conjugate method).
Why C Is Wrong: C results from computing only the denominator at x = 3: 3 – 3 = 0, then using 3 as the answer. The limit is not the denominator value.
Why D Is Wrong: D is correct.
Use the table below to understand your performance and plan your next study steps.
| Score | Performance Level | Next Step |
| 14–15 correct | Excellent — AP Ready | Proceed to Unit 2. Examine any incorrect responses. You are expected to receive a 4 or 5. |
| 11–13 correct | Good — Minor Gaps | Read the explanations for each incorrect response again. Determine which CED subtopics require review. |
| 7–10 correct | Developing — Review Needed | Reexamine IVT, limit selection techniques, discontinuity types, and continuity conditions. |
| 0–6 correct | Needs Full Re-Study | Before retesting, restart Unit 1 from 1.2 and watch the video lessons for each subtopic. |
These mistakes appear most frequently in College Board AP Calculus AB Unit 1 progress checks and the full AP exam, based on Chief Reader Reports and AP educator analysis.
| Mistake | Why It Loses Points | How to Fix It |
| Choosing DNE when getting 0/0 | DNE is not the same as 0/0, which is an indeterminate form. | Always factor, rationalize, or use conjugate multiplication before drawing a conclusion when you get 0/0. |
| Confusing f(c) with the limit | A limit is not the same as a function value. | Is the question asking for the VALUE (f(c)) or the LIMIT (behavior as x→c)? They may be different. |
| Failing the 3rd continuity condition | For continuity, limit exists + f(c) defined is insufficient. | Clearly state each of the three requirements. The limit cannot simply exist; it must EQUAL f(c). |
| Wrong rule for limits at infinity | Rather than comparing leading terms, students add all coefficients. | Only the highest degree terms should be used: equal degrees → ratio of leading coefficients. |
| Misidentifying IVT guarantees | IVT ensures that c exists, but not its location or monotonicity. | Just make sure that N is strictly between f(a) and f(b). If so, IVT ensures that for some c, f(c) = N. |
Avoiding these 5 mistakes can recover 3–5 points on the AP Calculus AB Unit 1 test. Review each one before your next practice session.
| Step | Action |
| Step 1 — Take the full test timed. | Set a timer for thirty minutes. Not a calculator. Before reviewing any responses, complete all 15 questions. |
| Step 2 — Score yourself with the answer key. | Prior to reading any explanations, mark each question as correct or incorrect. |
| Step 3 — Read ALL explanations. | Make sure your logic aligns with the explanation, even for right answers. Explanations of incorrect answers are particularly useful (UWorld method). |
| Step 4 — Map wrong answers to CED sub-topics. | Each question has its CED subtopic labeled. For every subtopic you overlooked, go back to the College Board CED. |
| Step 5 — Retake wrong questions next day. | Spaced repetition: don’t try missed questions again for a full day. Long-term retention is greatly enhanced by this. |
| Step 6 — Track your difficulty pattern. | If you tend to miss hard questions, concentrate on IVT (1.16), piecewise continuity (1.11), and the Squeeze Theorem (1.8). |
What is on the AP Calculus AB Unit 1 test?
Limits and Continuity are covered in 16 subtopics (1.1–1.16) of the College Board CED in the AP Calculus AB Unit 1 exam. The Squeeze Theorem, three types of discontinuity, continuity conditions, removing discontinuities, vertical and horizontal asymptotes, one-sided limits, evaluating limits algebraically and from graphs, and the Intermediate Value Theorem are important subjects.
How much of the AP Calculus AB exam is Unit 1?
Ten to twelve percent of the total score on the AP Calculus AB exam comes from Unit 1 (Limits and Continuity), which is equivalent to five to six multiple-choice questions. FRQ sections that call for IVT application or continuity justification also use Unit 1 concepts.
What are the three types of discontinuity in AP Calculus AB?
The three types are as follows: (1) Removable discontinuity, which leaves a hole in the graph because the limit exists but does not equal f(c). (2) Jump discontinuity: There are one-sided limits on the left and right, but they are not equal. (3) Infinite discontinuity: the graph has a vertical asymptote when the limit equals ±∞.
What is the Intermediate Value Theorem in AP Calculus AB?
According to the IVT, there must be at least one c in (a, b) where f(c) = N if f is continuous on a closed interval [a, b] and N is any value between f(a) and f(b). The existence of c is guaranteed by the IVT, but neither its location nor the number of such values are disclosed.
This resource is created for U.S. high school students by AP-certified educators at TestPrepKart. All sub-topic names, exam weights, and CED references are sourced from the official College Board AP Calculus AB Course and Exam Description. For the most current exam information, always visit apcentral.collegeboard.org. Last Updated: 2025.
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