AP Calculus ABAP Exams

AP Calculus AB Unit 1 Review: The Most-Tested Limits and Continuity Questions

Q: How much of the AP Calculus AB exam is limits and continuity?

The CED gives Unit 1 10–15% of the multiple choice. On the 7 official practice exams from 2012–2018 it was 33 of 315 multiple choice questions (10.5%), 4 or 5 per exam. In free response it is much smaller: 2.5% of the points on the 2012–2026 exams, usually one part of a larger question.

Q: Is L'Hospital's Rule part of Unit 1?

No. In the CED it is topic 4.7, in Unit 4. It is worth reviewing alongside limits, though: it was in 5 free-response questions since 2012, 4 of them since 2019.

Q: Is the squeeze theorem on the AP Calculus AB exam?

It is in the CED (topic 1.8), so it can be tested, but no official multiple choice question has been about it. One free-response part used the idea: 2019 Question 6(d) asked whether a function trapped between two others is continuous. Learn it, but give it less time than the main limit types.

Q: How do you justify an answer with the Intermediate Value Theorem on the free response?

State that the function is continuous on the closed interval (if it is given as differentiable, say that differentiable implies continuous), give the two endpoint values, and show that the target value lies between them. Then conclude that the IVT guarantees a value c in the interval where the function equals the target. IVT justifications were in 6 free-response questions from 2012 to 2026.

Q: Do I need a calculator for Unit 1 questions?

Mostly not. Of the 33 Unit 1 multiple choice questions on the 2012–2018 practice exams, 22 were in the no-calculator part. The 11 in the calculator part were reasoning questions, such as the IVT and what continuity implies, not limits you would compute on a calculator.

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Key takeaways

Unit weight: about a tenth of the multiple choice (33 of 315 official questions, 10.5%) and 2.5% of the free-response points.
Top types: limits at infinity, horizontal asymptotes, and what continuity, limits and differentiability imply.
Practice: writing an IVT justification. It was in 6 free-response questions since 2012, and multiple choice never asks for one.

Limits and continuity are about a tenth of the AP Calculus AB multiple choice, and much less of the free response. College Board’s Course and Exam Description (the CED) gives Unit 1 10–15% of the multiple choice. On the 7 official practice exams from 2012–2018 it was 33 of 315 questions (10.5%), 4 or 5 per exam, at the bottom of that range. On the 42-question section that starts in May 2027, that rate is about 4 or 5 questions.

In free response, Unit 1 earned just 2.5% of the points on the 2012–2026 exams. Limits rarely get a question of their own; they show up as one part, worth 1 or 2 points, inside a question about something else. Of the 86 free-response questions since 2012, 13 had a Unit 1 part: 6 Intermediate Value Theorem (IVT) justifications, 3 continuity checks at a point (such as where two pieces of a function meet), 3 limits at infinity, and 1 limit read from a graph.

Two limit skills students review here sit in other units. L’Hospital’s Rule is topic 4.7 (Unit 4) and was in 5 free-response questions since 2012, 4 of them since 2019. Recognizing a limit as a derivative is Unit 2 and was on all 7 practice exams. Review both with this unit.

The 10 most-tested Unit 1 question types

Ranked by share of the whole exam. Each is about 1% or less, and the ten add up to about 6%: quick points if you know the moves.

1. Limits at infinity

Compare how fast the top and bottom grow as $x \to \pm\infty$ .

Topic 1.151.0% of the exam4 of 7 practice exams2 FRQs since 2012

Try it

Find

\lim\limits_{x \to \infty} \frac{2x^2 + 7}{5x^3 - x}

Solution

0. The bottom has the higher degree.

Common mistake

dividing the leading coefficients when the degrees differ. A lower-degree top gives 0; a higher-degree top gives an unbounded quotient.

Practice: Limits at Infinity worksheet · AP Unit 1 worksheet

2. Horizontal asymptotes

Find every horizontal asymptote: check both ends.

Topic 1.150.8% of the exam4 of 7 practice exams1 FRQ since 2012

Try it

Find the horizontal asymptotes of

f(x) = \frac{3x}{\sqrt{x^2 + 4}}

Solution

$y = 3$ and $y = -3$ . $\sqrt{x^2} = |x|$ , which is $-x$ for negative $x$ .

Common mistake

finding the asymptote at only one end, or reading it off the leading terms when the degrees differ.

Practice: Limits at Infinity worksheet · AP Unit 1 worksheet

3. What continuity, limits and differentiability imply

Decide what must be true when $f$ is continuous, has a limit, or is differentiable at a point. On free response: “Is $g$ continuous at $x = a$ ? Justify,” answered with both one-sided limits and the value.

Topic 1.110.8% of the exam3 of 7 practice exams3 FRQs since 2012

Try it

f

is continuous at

x = 2

and

f(2) = 5

. Which must be true?

I. $\lim\limits_{x \to 2} f(x) = 5$ .
II. $f'(2)$ exists.
III.The left- and right-hand limits at 2 are equal.

Solution

I and III only. $f(x) = |x - 2| + 5$ is continuous at 2 but has no derivative there.

Common mistake

reversing an implication. Differentiable implies continuous, and continuous implies the limit exists, but none of these arrows goes back.

Practice: Continuity and Discontinuities worksheet · AP Unit 1 worksheet

4. Limits from a graph

Read one- and two-sided limits and values off a graph with open and filled dots.

Topic 1.30.7% of the exam5 of 7 practice exams, the most of any Unit 1 typeno FRQs since 2012

Try it

The graph of

g

is shown.

Find

\lim\limits_{x \to 1} g(x)

and

g(1)

Solution

the limit is 3, and $g(1) = -2$ .

Common mistake

giving the filled dot’s value as the limit, or deciding a limit does not exist because of an open dot.

Practice: Limits From a Graph worksheet · AP Unit 1 worksheet

5. Locate and classify discontinuities

Find where $f$ breaks and name the kind: removable, jump or vertical asymptote.

Topic 1.100.6% of the exam3 of 7 practice examsno FRQs since 2012

Try it

Where is

f(x) = \frac{x^2 - 9}{x^2 - x - 6}

discontinuous, and what kind?

Solution

$f(x) = \frac{x + 3}{x + 2}$ for $x \ne 3$ , so removable at $x = 3$ (limit $\frac{6}{5}$ ) and a vertical asymptote at $x = -2$ .

Common mistake

missing the removable discontinuity where a factor cancels.

f

is still undefined there.

Practice: Continuity and Discontinuities worksheet · AP Unit 1 worksheet

6. Find the constant that makes $f$ continuous

Set the pieces equal where they meet and solve.

Topic 1.130.6% of the exam3 of 7 practice examsno FRQs since 2012

Try it

f(x) = kx + 1

for

x < 2

and

x^2 - k

for

x \ge 2

. Find

k

so that

f

is continuous.

Solution

$2k + 1 = 4 - k$ , so $k = 1$ .

Common mistake

setting the slopes of the pieces equal, which is the condition for differentiability, instead of their values.

Practice: Continuity of Piecewise Functions worksheet · AP Unit 1 worksheet

7. Limits by factoring or rationalizing (0/0)

Substitution gives $\frac{0}{0}$ , so factor and cancel, or use a conjugate.

Topic 1.60.6% of the exam2 of 7 practice examsno FRQs since 2012

Try it

Find

\lim\limits_{x \to 4} \frac{\sqrt{x + 5} - 3}{x - 4}

Solution

multiplying by the conjugate leaves $\frac{1}{\sqrt{x + 5} + 3}$ , which goes to $\frac{1}{6}$ .

Common mistake

getting

\frac{0}{0}

and answering “does not exist” instead of simplifying.

Practice: Limits at Removable Discontinuities worksheet · AP Unit 1 worksheet

8. Justify with the IVT (free response)

Explain why a function must take a value on an interval, usually from a table.

Topic 1.160.5% of the examno multiple choice version6 FRQs since 2012, the most of any Unit 1 type

Try it

h

is continuous,

h(1) = -2

and

h(4) = 7

. Must

h(c) = 3

for some

c

between 1 and 4?

Solution

yes. $h$ is continuous on $[1, 4]$ and $-2 < 3 < 7$ , so the IVT guarantees such a $c$ .

Common mistake

writing “by the IVT” without stating that the function is continuous on the interval and that the value lies between the two endpoint values. If the question says the function is differentiable, say so: differentiable implies continuous.

Practice: Intermediate Value Theorem worksheet · AP Unit 1 worksheet

9. The IVT in multiple choice

Pick the interval that must contain a zero or a value.

Topic 1.160.4% of the exam3 of 7 practice examsfree-response use is counted under #8

Try it

f(x) = x^3 + x - 5

. Which interval must contain a zero of

f

$(0, 1)$
$(1, 2)$
$(2, 3)$
$(3, 4)$

Solution

(B) $(1, 2)$ . $f$ is continuous and $f(1) = -3 < 0 < 5 = f(2)$ .

Common mistake

using the IVT without continuity on the closed interval, or expecting it to say where the value is taken.

Practice: Intermediate Value Theorem worksheet · AP Unit 1 worksheet

10. Limits of piecewise functions and combinations

One-sided limits of a piecewise function, or limits of sums, products and compositions.

Topic 1.50.4% of the exam2 of 7 practice exams1 FRQ since 2012

Try it

f(x) = x + 1

for

x < 1

and

x - 3

for

x \ge 1

. Does

\lim\limits_{x \to 1} f(x)

exist? Does

\lim\limits_{x \to 1} [f(x)]^2

Solution

no, the one-sided limits are 2 and $-2$ . Yes, both one-sided limits of $[f(x)]^2$ are 4.

Common mistake

using the piece that contains the break point for the limit from the other side, or deciding a combination has no limit because one part has none.

Practice: One-Sided Limits worksheet · AP Unit 1 worksheet

What’s rising and fading in Unit 1

We compare each type’s share of recent exams (multiple choice 2012–2018, free response 2012–2026) with its share across all years: Rising is at least double, Fading under half, Dormant not seen since 2011. Most of Unit 1 is Steady. The exceptions:

Rising: IVT justifications on free response. 0.69% of recent exams against 0.31% across all years; asked in 2014, 2016, 2020, 2022, 2025 and 2026.
Labeled Rising, but on one question: limits from a table. Its only official question is from 2018: it is on the exam now, but that is not a trend.
Fading: infinite limits and vertical asymptotes. 0.16% against 0.36%: one practice-exam question (2015), no free-response use since 2008.
Dormant: special trig limits such as $\frac{\sin x}{x}$ . Two official questions ever, the last in 1993.
Just outside Unit 1: L’Hospital’s Rule is Rising. 1.17% against 0.40%, and its free-response share went from 0.8% of points (2012–2018) to 2.6% (2019–2026).

Unit 1 as a whole went from 1.6% of free-response points (2012–2018) to 3.4% (2019–2026). One question is about 2.3% of a window’s points, so that is about one extra question, not a shift.

In the CED, rarely or never tested

Still testable, so don’t skip them, but lower priority than the ten above.

The squeeze theorem (topic 1.8). No official multiple choice question, not even as a secondary topic. One free-response part used the idea: 2019 Question 6(d) asked whether a function trapped between two others is continuous.
Continuity over an interval (topic 1.12). No official multiple choice question targets it, but it is the hypothesis behind every IVT justification.
Topics 1.1, 1.2, 1.7 and 1.9 (rates at an instant, limit notation, choosing a limit method, moving between graphs, tables and formulas). None is the main topic of an official question; every limit question uses them.
The epsilon-delta definition of a limit is not assessed, per the CED.

What students over- and under-practice in Unit 1

We tagged 10,263 multiple choice questions from teacher worksheets, review packets and prep books (reposted official questions and non-AB content removed). Unit 1 is 12.0% of them, against 9.0% of the 719 in-scope official multiple choice questions from 1969–2018. Per type:

Question type	Practice vs official share	Ratio
Special trig limits	0.78% vs 0.28%	2.8× over
Limits of piecewise functions and combinations	1.17% vs 0.42%	2.8× over
Infinite limits and vertical asymptotes	0.74% vs 0.28%	2.7× over
Limits by factoring (0/0)	1.65% vs 0.70%	2.4× over
Locate and classify discontinuities	1.48% vs 0.70%	2.1× over
L’Hospital’s Rule (Unit 4)	0.84% vs 0.42%	2.0× over
Limits at infinity	1.97% vs 1.25%	1.6× over
Horizontal asymptotes	0.80% vs 1.53%	0.5× under
What continuity, limits and differentiability imply	0.62% vs 0.97%	0.6× under
The IVT in multiple choice	0.67% vs 0.83%	0.8× under

Official counts per type are small (2 to 11 questions for most), so read the ratios as directions.

Algebra-style limits are drilled at 2–3 times their exam share. Keep a few of each; don’t spend a week on them.
The reasoning questions get too little practice: asymptotes at both ends, and what continuity does and doesn’t imply.
The biggest gap has no multiple choice version. IVT justifications were in 12 free-response questions since 1998; none of the practice questions we collected asks you to write one.
Graphs and tables are about even: limits from a graph 0.9×, from a table 1.0× (15 practice questions against 1 official).

Calculator note

Of the 33 Unit 1 multiple choice questions on the 2012–2018 practice exams, 22 were in the no-calculator part and 11 in the calculator part. The calculator-part ones were reasoning, not computing: all 3 IVT questions, all 3 “what does continuity imply” questions, 2 of the 5 limits from a graph, the one limit from a table, 1 of the 5 horizontal asymptote questions and the one infinite-limit question. No limit at infinity, factoring, discontinuity, piecewise or find-the-constant question was. On free response, 3 of the 6 IVT justifications were in calculator questions (2016, 2020 and 2026; the 2020 at-home exam allowed a calculator throughout).

So compute Unit 1 limits by hand, and expect IVT and continuity questions in either part.

How we counted

Every number here comes from the public official AP Calculus AB exams: 760 multiple choice questions (1969–2018) and 416 free-response questions (1969–2026), each tagged to its CED topic and question type. Every example problem is our own.

Practice Unit 1

Our free AP Calculus AB Unit 1 worksheet has 20 original multiple choice questions, a no-calculator Part A and a calculator Part B, weighted by how often the official exams ask each type. It comes with an answer key and a printable PDF.

For free response, the AP Calculus AB Unit 1 FRQ set has 4 multi-part questions in the exam’s form, each scored out of 9 points, with a scoring guide and a PDF. Both make good practice before the Unit 1 Progress Check in AP Classroom.

Common Questions

The CED gives Unit 1 10–15% of the multiple choice. On the 7 official practice exams from 2012–2018 it was 33 of 315 multiple choice questions (10.5%), 4 or 5 per exam. In free response it is much smaller: 2.5% of the points on the 2012–2026 exams, usually one part of a larger question.

No. In the CED it is topic 4.7, in Unit 4. It is worth reviewing alongside limits, though: it was in 5 free-response questions since 2012, 4 of them since 2019.

It is in the CED (topic 1.8), so it can be tested, but no official multiple choice question has been about it. One free-response part used the idea: 2019 Question 6(d) asked whether a function trapped between two others is continuous. Learn it, but give it less time than the main limit types.

State that the function is continuous on the closed interval (if it is given as differentiable, say that differentiable implies continuous), give the two endpoint values, and show that the target value lies between them. Then conclude that the IVT guarantees a value c in the interval where the function equals the target. IVT justifications were in 6 free-response questions from 2012 to 2026.

Mostly not. Of the 33 Unit 1 multiple choice questions on the 2012–2018 practice exams, 22 were in the no-calculator part. The 11 in the calculator part were reasoning questions, such as the IVT and what continuity implies, not limits you would compute on a calculator.

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AP Calculus AB Unit 1 Review: The Most-Tested Limits and Continuity Questions

The 10 most-tested Unit 1 question types

1. Limits at infinity

2. Horizontal asymptotes

3. What continuity, limits and differentiability imply

4. Limits from a graph

5. Locate and classify discontinuities

6. Find the constant that makes fff continuous

7. Limits by factoring or rationalizing (0/0)

8. Justify with the IVT (free response)

9. The IVT in multiple choice

10. Limits of piecewise functions and combinations

What’s rising and fading in Unit 1

In the CED, rarely or never tested

What students over- and under-practice in Unit 1

Calculator note

How we counted

Practice Unit 1

Common Questions

How much of the AP Calculus AB exam is limits and continuity?+

Is L'Hospital's Rule part of Unit 1?+

Is the squeeze theorem on the AP Calculus AB exam?+

How do you justify an answer with the Intermediate Value Theorem on the free response?+

Do I need a calculator for Unit 1 questions?+

Practice this unit: free AP Calculus AB Unit 1 worksheet

Keep reading

AP Calculus AB Review: The Most-Tested Questions, Unit by Unit

AP Calculus AB Unit 2 Review: The Most-Tested Derivative Questions

AP Calculus AB Unit 3 Review: The Most-Tested Chain Rule and Implicit Questions

6. Find the constant that makes $f$ continuous