Worksheets · AP Calculus AB

AP Calculus AB Unit 1 Worksheet: Limits and Continuity

20 exam-style multiple-choice questions on limits and continuity, free to use in class. Good practice before the Unit 1 Progress Check in AP Classroom. Free response for this unit: Unit 1 FRQ set.

Sigma Prep · sigmaprep.io/worksheets/ap-calculus-ab

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AP Calculus AB Unit 1: Limits and Continuity

Date Version A

Directions: Work each problem in the space below it, then choose the best of the four answer choices. Unless a question says otherwise, the domain of a function f is every real number x for which f(x) is a real number.

Part A: No calculator is allowed.

Questions 1–13
  1. For which function below is the line y=3 a horizontal asymptote of its graph?
    1. (A)y=3xx2+3
    2. (B)y=12x2−x4x2+3
    3. (C)y=3x
    4. (D)y=sin⁡(3x)x
  2. The function f is defined by f(x)={−x−2for x≤−1−4x−5for x>−1. Which of the following describes the graph of f at x=−1?
    1. (A)A jump discontinuity
    2. (B)A vertical asymptote
    3. (C)A removable discontinuity
    4. (D)No discontinuity; f is continuous there
  3. A function is defined in two pieces: f(x)={2x−5for x<4x−8for x≥4.
    What is lim⁡x→4−f(x)?
    1. (A)3
    2. (B)−4
    3. (C)nonexistent
    4. (D)−12
  4. f(x)=(x−1)(x−1)(x−3)List every value of x where the function above fails to be continuous.
    1. (A)3 only
    2. (B)−3 and −1 only
    3. (C)1 only
    4. (D)1 and 3 only
  5. f(x)={kx−7for x≤−13x2−5for x>−1.
    If k is a constant and f is continuous at x=−1, then k=
    1. (A)5
    2. (B)−6
    3. (C)−15
    4. (D)−5
  6. Find all lines y=L that the graph of y=2x+63−2x approaches as x→∞ or as x→−∞.
    1. (A)y=−1 only
    2. (B)y=−1 and y=2
    3. (C)y=1 and y=2
    4. (D)y=0 and y=2
  7. What number is equal to the limit lim⁡x→−6x2+x−30x2+3x−18?
    1. (A)119
    2. (B)911
    3. (C)1
    4. (D)0
  8. Which of the following is equal to the limit lim⁡x→−∞−9−5ex8+4ex?
    1. (A)−54
    2. (B)−98
    3. (C)−94
    4. (D)0
  9. The figure shows the graph of a function f, made of line segments. Only one of the following agrees with the graph. Which is it?
    −3−2−10123456789101112−2−1012345xyy = f(x)
    1. (A)f is continuous at x=5
    2. (B)f is continuous at x=3
    3. (C)f(7)=2
    4. (D)lim⁡x→7f(x) exists
  10. The limit lim⁡x→∞2x3−5x2+6x−4−6x2−2x+3 is equal to which of the following?
    1. (A)−∞
    2. (B)∞
    3. (C)−13
    4. (D)0
  11. f(x)={kfor x=0sin⁡(6x)3xfor x≠0.
    For this function to be continuous at x=0, what must the constant k equal?
    1. (A)0
    2. (B)12
    3. (C)13
    4. (D)2
  12. Which of the following gives lim⁡x→−∞−5x+9x2+3?
    1. (A)nonexistent
    2. (B)5
    3. (C)−5
    4. (D)0
  13. Which of the following gives lim⁡x→01−cos⁡2(7x)9x2?
    1. (A)19
    2. (B)4918
    3. (C)499
    4. (D)nonexistent

Part B: A graphing calculator is allowed.

Questions 14–20
  1. Let f(x)=ex/3+5sin⁡x, a function that is continuous for all real x. On which of the following intervals does the Intermediate Value Theorem guarantee a solution of f(x)=0.5?
    1. (A)[3,4]
    2. (B)[1,2]
    3. (C)[2,3]
    4. (D)[0,1]
  2. For f(x)={x2+3x−1for x≤13xfor x>1, which of the statements below hold at x=1?
    I. lim⁡x→1+f(x)=f(1)
    II. f is continuous at x=1.
    III. f is differentiable at x=1.
    1. (A)II and III only
    2. (B)I, II, and III
    3. (C)I and II only
    4. (D)I only
  3. A twice-differentiable function R gives the rate at which water flows through a drainage pipe, in liters per minute, at time t minutes after a study began (0≤t≤9). Measurements of R(t) are listed in the table.
    t (minutes) 04679R(t) (L/min) 78103804583Does the table guarantee that R(c)=50 for some time c strictly between 0 and 9? Which response gives the right conclusion with a valid reason?
    1. (A)Yes, because 50 is between R(7)=45 and R(9)=83.
    2. (B)No, because no value in the table equals 50.
    3. (C)Yes, because R is continuous on [7,9] and 50 is between R(7)=45 and R(9)=83.
    4. (D)Yes, by the Mean Value Theorem, because R is twice-differentiable on [7,9].
  4. x−0.1−0.01−0.0010.0010.010.1f(x)0.51320.50130.50010.49990.49880.4881The table shows values of a function f for x near 0. Based on the table, what is the most reasonable estimate of lim⁡x→0f(x)?
    1. (A)0.5001
    2. (B)0.4881
    3. (C)0.5
    4. (D)nonexistent
  5. The graph of f approaches the line x=−2 as a vertical asymptote. Which statement below is impossible for this function?
    1. (A)lim⁡x→−2f(x)=5
    2. (B)lim⁡x→−2+f(x)=−∞
    3. (C)f(−2)=5
    4. (D)lim⁡x→−2−f(x)=5
  6. Based on the graph of f in the figure, what is lim⁡x→5−f(x)?
    −9−8−7−6−5−4−3−2−10123456789−5−4−3−2−1012345678xyy = f(x)
    1. (A)2
    2. (B)0
    3. (C)−1
    4. (D)nonexistent
  7. About a function f, the only thing known is that lim⁡x→4f(x)=3. Which of the following statements are guaranteed?
    I. f(4)=3
    II. lim⁡x→4f(x) exists.
    III. f′(4)=−3
    1. (A)I and II only
    2. (B)II and III only
    3. (C)I, II, and III
    4. (D)II only

    End of Version A

Answer key · Version A

12345678910
BDADDBABBA
1.151.101.51.101.131.151.61.151.31.15
11121314151617181920
DBCACCCACD
1.131.151.61.161.111.161.41.141.31.11

The second row is the correct letter; the third is the course topic each question tests.

Free AP Calculus AB worksheets at sigmaprep.io/worksheets/ap-calculus-abOriginal questions written by Sigma Prep. AP is a registered trademark of the College Board, which is not affiliated with this sheet.

What does the AP Calculus AB exam ask on Unit 1?

About a tenth of the multiple choice: Unit 1 was 33 of the 315 official multiple-choice questions on the seven 2012–2018 practice exams (10.5%).

This worksheet has 20 questions in that mix: Part A (13) without a calculator and Part B (7) with one; the exam put 31% of its official Unit 1 multiple choice in the calculator section.

Counted from the official AP Calculus AB exams: multiple choice from 17 exams, 1969–2018, free response 2012–2026. No official question is reproduced here.

AP Calculus AB Unit 1 question types: how often the exam asks each

Official AP Calculus AB multiple-choice questions by question type, Unit 1 (Limits and Continuity), and how many are on this worksheet
Question typeOn this worksheetOfficial questions, 1969–2018Exams it was on (of 17)On the 2012–2018 practice exams
Limits from a graphTopic 1.32885
Limits from a tableTopic 1.41111
Limits of piecewise functions and combinationsTopic 1.51332
Limits by factoring (0/0)Topic 1.61552
Special trig limitsTopic 1.61220
Locate and classify discontinuitiesTopic 1.102553
What limits, continuity and differentiability implyTopic 1.112773
Find the value or constant that makes f continuousTopic 1.132663
Infinite limits and vertical asymptotesTopic 1.141221
Limits at infinityTopic 1.153985
Horizontal asymptotesTopic 1.15211105
Intermediate Value TheoremTopic 1.161653
Justify with the IVT (FRQ)Topic 1.161Asked on 12 free-response questions since 1998, most recently in 2026.

Counts are of official multiple-choice questions whose main topic is that kind, on 17 College Board exams 1969–2018 (7 released 1969–1998, 2003 and 2008 released, the 2008 practice exam and the seven 2012–2018 practice exams). No official multiple-choice questions after 2018 are public. Left out: average vs instantaneous rate of change; write or interpret limit notation; choose a procedure for a limit; limits by the squeeze theorem; limits across graphs, tables, formulas and words; continuity over an interval.

Free-response practice for this unit: 4 FRQs with a scoring guide.

Why these kinds of question: AP Calculus AB Unit 1 Review: The Most-Tested Limits and Continuity Questions.

Questions about this worksheet

Unit 1, Limits and Continuity, has 16 topics in the College Board's course description: limits from graphs, tables and formulas; one-sided and infinite limits; limits at infinity and asymptotes; algebraic methods such as factoring and the special trig limits; the squeeze theorem; types of discontinuities; continuity at a point and over an interval; and the Intermediate Value Theorem. The course description weights the unit at 10–15% of the multiple-choice section.

Only in Part B, as on the exam. Questions 1–13 are Part A, with no calculator; questions 14–20 are Part B, where the exam allows a graphing calculator (on the digital exam, the Desmos calculator built into Bluebook is available there). None of these Unit 1 questions needs one: Part B holds the kinds of questions the exam has placed in its calculator section, such as what continuity implies and the Intermediate Value Theorem. From May 2027 the exam's multiple-choice section is 29 questions without a calculator and 13 with one.

We tagged 760 official multiple-choice questions from 17 College Board exams (1969–2018) and the free-response questions from 1969 to 2026, sorted the Unit 1 questions into 19 kinds, and weighted each kind by how often and how recently the exam asks it. Every kind the exam actually uses gets at least one question, and the most-tested get more (3 on limits at infinity, 2 on limits from a graph, 2 on locate and classify discontinuities, 2 on what limits, continuity and differentiability imply, 2 on find the value or constant that makes f continuous, 2 on horizontal asymptotes). The 6 kinds that have never been the main topic of an official multiple-choice question are left out.

The progress check is College Board's own end-of-unit quiz in AP Classroom, assigned by your teacher. Its questions are not public, and we do not have them: Sigma Prep is not affiliated with College Board. This worksheet is exam-style practice on the same unit: 20 original multiple-choice questions, weighted by how often the official exams ask each kind of question, with a no-calculator Part A and a calculator Part B. Time yourself, check the answer key, then redo the kinds you missed. For the free-response part, use the Unit 1 FRQ set, which has a scoring guide.

No. Every question here is original, written by Sigma Prep in the style of the exam. We studied which kinds of questions the official exams ask and how often; no official question is copied or reworded. Each answer was checked by an independent computer-algebra check before the sheet was published.

Yes. It is shown with the sheet, and the PDF prints it on its own last page. There is also a PDF without the key to hand out. Worked solutions are not included.

AP is a registered trademark of the College Board, which was not involved in making this worksheet and does not endorse it. All questions are original to Sigma Prep. Frozen 2026-10-24.