Limits and Continuity Practice Questions
AP (Advanced Placement) · AP Calculus AB · 139 free MCQs with instant results and detailed explanations.
139
Total
39
Easy
68
Medium
32
Hard
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Sample Questions from Limits and Continuity
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Q1
Easy
What is the limit of f(x) = 3x + 2 as x approaches 4?
A.
14
B.
12
C.
10
D.
8
Show Answer & Explanation
Correct Answer: A
To find the limit as x approaches 4, substitute x = 4 into the function: f(4) = 3(4) + 2 = 12 + 2 = 14.
Q2
Easy
What is the limit of the function h(x) = sin(x)/x as x approaches 0?
A.
0
B.
1
C.
undefined
D.
Infinity
Show Answer & Explanation
Correct Answer: B
The limit of sin(x)/x as x approaches 0 is a well-known limit, which equals 1.
Q3
Easy
Determine the continuity of the function g(x) = 1/(x-2) at x = 2.
A.
Continuous
B.
Discontinuous
C.
Only left continuous
D.
Only right continuous
Show Answer & Explanation
Correct Answer: B
The function g(x) = 1/(x-2) is discontinuous at x = 2 because the function is undefined at this point, leading to a vertical asymptote.
Q4
Medium
Evaluate the limit: lim(x→3) (x^2 - 9)/(x - 3)
A.
6
B.
3
C.
0
D.
undefined
Show Answer & Explanation
Correct Answer: A
The expression (x^2 - 9) can be factored as (x - 3)(x + 3). Cancelling (x - 3) gives us lim(x→3) (x + 3) = 6.
Q5
Medium
Which of the following statements is true about the limit: lim(x→0) (sin(x)/x)?
A.
The limit equals 0.
B.
The limit equals 1.
C.
The limit is undefined.
D.
The limit does not exist.
Show Answer & Explanation
Correct Answer: B
The limit lim(x→0) (sin(x)/x) is a standard result and equals 1 due to the Squeeze Theorem or L'Hospital's rule.
Q6
Medium
If f(x) = 2x^2 - 4x + 1, what is the value of the limit as x approaches 2?
A.
1
B.
0
C.
3
D.
undefined
Show Answer & Explanation
Correct Answer: C
Substituting x = 2 into f(x) gives f(2) = 2(2^2) - 4(2) + 1 = 3.
Q7
Medium
Determine the continuity of the function f(x) = { x^2 if x < 1, 2 if x = 1, x + 1 if x > 1 } at x = 1.
A.
The function is continuous at x = 1.
B.
The function is not continuous at x = 1.
C.
The function is piecewise continuous.
D.
The function has a removable discontinuity at x = 1.
Show Answer & Explanation
Correct Answer: B
The limit from the left (1) does not equal the value at f(1) (2), hence the function is discontinuous at x = 1.
Q8
Hard
Evaluate the limit: lim (x -> 0) [(sin(5x) - 5x) / x^3]. What is the value of this limit?
A.
0
B.
25/6
C.
-25/6
D.
5
Show Answer & Explanation
Correct Answer: B
Using L'Hôpital's rule three times on the limit, we calculate the derivatives until we reach a non-indeterminate form, yielding the result 25/6.
Q9
Hard
Determine the continuity of the function f(x) = { x^2 sin(1/x) for x ≠ 0; 0 for x = 0 }. Is f(x) continuous at x = 0?
A.
Yes, it is continuous.
B.
No, it has a removable discontinuity.
C.
No, it is discontinuous.
D.
Yes, but only differentiable elsewhere.
Show Answer & Explanation
Correct Answer: A
The limit of f(x) as x approaches 0 equals f(0) = 0, satisfying the continuity condition.
Q10
Hard
Evaluate the limit: lim (x → 0) (sin(5x)/x)
A.
5
B.
1
C.
0
D.
Undefined
Show Answer & Explanation
Correct Answer: A
The limit of sin(kx)/x as x approaches 0 is k, where k is a constant. In this case, k = 5, thus the limit evaluates to 5.
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Limits and Continuity — AP (Advanced Placement) AP Calculus AB Practice Questions Online
This page contains 139 practice MCQs for the chapter Limits and Continuity in AP (Advanced Placement) AP Calculus AB. The questions are organized by difficulty — 39 easy, 68 medium, 32 hard — so you can choose the right level for your preparation.
Every question includes a detailed explanation to help you understand the concept, not just memorize answers. Take a timed quiz to simulate exam conditions, or practice at your own pace with no time limit.