AP Calculus AB 1 — Questions and Answers
Question 1: What is lim(x→3) (x² − 9)/(x − 3)?
- 0
- 3
- 6 (Correct answer)
- Undefined
Correct answer: 6
Factor the numerator as (x+3)(x−3); canceling (x−3) leaves x+3, which equals 6 at x=3.
Question 2: Which condition is NOT required for a function f(x) to be continuous at x = a?
- f(a) is defined
- lim(x→a) f(x) exists
- lim(x→a) f(x) = f(a)
- f′(a) exists (Correct answer)
Correct answer: f′(a) exists
Differentiability implies continuity, but continuity does not require differentiability; only the three limit/value conditions are needed.
Question 3: What is lim(x→0) sin(x)/x?
- 0
- 1 (Correct answer)
- ∞
- Does not exist
Correct answer: 1
This is a fundamental trigonometric limit — sin(x)/x approaches 1 as x approaches 0.
Question 4: If lim(x→2⁺) f(x) = 5 and lim(x→2⁻) f(x) = 3, what is lim(x→2) f(x)?
- 5
- 3
- 4
- The limit does not exist (Correct answer)
Correct answer: The limit does not exist
A two-sided limit exists only when both one-sided limits are equal; since 5 ≠ 3, the limit does not exist.
Question 5: What is lim(x→∞) (3x² + 2x)/(x² − 5)?
- 0
- 2
- 3 (Correct answer)
- ∞
Correct answer: 3
Dividing numerator and denominator by x² causes the lower-degree terms to vanish, leaving 3/1 = 3.
Question 6: Which function has a removable discontinuity at x = 2?
- f(x) = 1/(x − 2)
- f(x) = (x² − 4)/(x − 2) (Correct answer)
- f(x) = |x − 2|
- f(x) = √(x − 2)
Correct answer: f(x) = (x² − 4)/(x − 2)
Factoring gives (x+2)(x−2)/(x−2) = x+2 for x ≠ 2, leaving a hole (removable discontinuity) at x = 2.
Question 7: What is lim(x→0) (1 − cos x)/x²?
- 0
- 1/2 (Correct answer)
- 1
- 2
Correct answer: 1/2
Applying L'Hôpital's rule twice (or using the half-angle identity) yields 1/2.
AP Calculus AB 1
What is lim(x→3) (x² − 9)/(x − 3)?