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Limits and continuity practice problems AP Calculus, Unit 1

Written by STEM-Tutor's tutors · Updated October 2026 · 8 questions and 1 free-response question

Limits from graphs, tables and algebra, limits at infinity, continuity and the intermediate value theorem. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.

On this page
  1. Key equations
  2. Warm-up questions
  3. Exam-style questions
  4. Free-response practice
  5. Common mistakes
  6. Keep going

Key equations for limits and continuity

limx→a f(x) = Lf(x) gets as close as we like to L as x approaches a
limx→0 sin x / x = 1the key trigonometric limit
f continuous at af(a) exists, the limit exists, and they are equal
Intermediate value theoremcontinuous f on [a, b] takes every value between f(a) and f(b)

Questions marked No calculator should be done by hand, as in Part A of the multiple-choice section. You can type fractions, such as 3/4. The full sheet is in our AP Calculus formula sheet.

Warm-up questions

Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.

Q1Warm-upNo calculator
Find limx→3 (x2 − 9)/(x − 3).

Hint

Direct substitution gives 0/0. Factorise the top.

Worked solution
  1. Substituting x = 3 gives 0/0, so simplify first.
  2. x2 − 9 = (x − 3)(x + 3), so the fraction is x + 3 for x ≠ 3.
  3. The limit is 3 + 3 = 6.
Q2Warm-upNo calculator
A function f is continuous at x = a when:

Hint

Continuity needs three things to hold together.

Worked solution
  1. f(a) must exist, the limit must exist, and they must be equal.
  2. Option C includes all three conditions.
  3. Differentiability is stronger than continuity, so it is not required. The answer is C.
Q3Warm-upNo calculator
Find limx→∞ (3x2 + 1)/(5x2 − x).

Hint

Compare the highest powers on the top and bottom.

Worked solution
  1. Both top and bottom have degree 2.
  2. Divide every term by x2: (3 + 1/x2)/(5 − 1/x) → 3/5.
  3. The limit is 3/5 = 0.6.

Exam-style questions

Questions at the level of the AP Calculus exam. The ones marked Challenge combine more than one idea.

Q4Exam-styleNo calculator
Find limx→0 sin(5x)/x.

Hint

Make the bottom match the angle inside the sine.

Worked solution
  1. Write sin(5x)/x = 5 × sin(5x)/(5x).
  2. As x → 0, 5x → 0, so sin(5x)/(5x) → 1.
  3. The limit is 5.
Q5Exam-styleNo calculator
f(x) = x2 + k for x < 2, and f(x) = 3x − 1 for x ≥ 2. What value of k makes f continuous at x = 2?

Hint

The two pieces must meet at x = 2.

Worked solution
  1. From the left: 22 + k = 4 + k.
  2. From the right, and f(2): 3(2) − 1 = 5.
  3. Set them equal: 4 + k = 5, so k = 1.
Q6Exam-styleNo calculator
Find limx→4 (√x − 2)/(x − 4). You may give a fraction.

Hint

Multiply the top and bottom by the conjugate, √x + 2.

Worked solution
  1. Substitution gives 0/0.
  2. (√x − 2)(√x + 2) = x − 4, so the fraction becomes 1/(√x + 2).
  3. At x = 4: 1/(2 + 2) = 1/4.
Q7Exam-styleNo calculator
f is continuous on [1, 3], with f(1) = −2 and f(3) = 5. Which must be true?

Hint

Which theorem guarantees a value between f(1) and f(3)?

Worked solution
  1. f is continuous, and 0 lies between f(1) = −2 and f(3) = 5.
  2. By the intermediate value theorem, f(c) = 0 for some c in (1, 3).
  3. The other options need more information. The answer is A.
Q8ChallengeNo calculator
Find limx→0 (1 − cos x)/x2.

Hint

Multiply the top and bottom by 1 + cos x, or use L'Hôpital's rule twice.

Worked solution
  1. Multiply by (1 + cos x)/(1 + cos x): the top becomes 1 − cos2x = sin2x.
  2. So the expression is (sin x / x)2 × 1/(1 + cos x).
  3. As x → 0 this tends to 12 × 1/2 = 1/2.

Free-response practice

One original question in the style of the exam's free-response section. Answer it in full, with working, before you open the scoring guide.

No-calculator free-response style · 6 points · about 15 minutes

Let g(x) = (x2 − 4)/(x − 2) for x ≠ 2, and g(2) = k.

  1. (a) Find limx→2 g(x). Show your work.
  2. (b) For what value of k is g continuous at x = 2? Justify your answer using the definition of continuity.
  3. (c) Let h(x) = (2x2 + 3)/(x2 − 1). Find limx→∞ h(x) and state what it tells you about the graph of h.

Write your full answer on paper first, then open the scoring guide and tick each point you earned.

Scoring guide and model answer

Part (a)

For x ≠ 2, g(x) = (x − 2)(x + 2)/(x − 2) = x + 2, so the limit is 4.

Part (b)

g is continuous at 2 when g(2) = limx→2 g(x). The limit is 4, so k = 4. Then g(2) exists, the limit exists, and they are equal.

Part (c)

Dividing by x2, the limit is 2. So y = 2 is a horizontal asymptote of h.

Your mark: 0 of 6

Common mistakes in limits and continuity

Keep going

Revise the ideas behind these questions:

Then try real exam questions: AP Calculus past papers, with official scoring guidelines. · All AP Calculus units