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Limits and Continuity Practice Questions

AP (Advanced Placement) · AP Calculus BC · 136 free MCQs with instant results and detailed explanations.

136
Total
39
Easy
69
Medium
28
Hard

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Sample Questions from Limits and Continuity

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Q1
Easy
If lim x->2 (x^2 - 4)/(x - 2) exists, what is its value?
A. 4
B. 2
C. 0
D. undefined
Show Answer & Explanation
Correct Answer: A
The limit can be simplified by factoring: (x^2 - 4) = (x - 2)(x + 2). Cancelling (x - 2) gives lim x->2 (x + 2) = 4.
Q2
Easy
Which of the following statements about continuity is true?
A. A function can be continuous at a point where it is undefined.
B. A function is continuous if it has a hole at a point.
C. A function is continuous at a point if the limit exists and equals the function's value at that point.
D. A function is continuous everywhere if it has vertical asymptotes.
Show Answer & Explanation
Correct Answer: C
A function is defined as continuous at a point if the limit at that point exists and equals the value of the function at that point.
Q3
Easy
What is the limit of f(x) = 2x + 3 as x approaches 4?
A. 11
B. 8
C. 7
D. 12
Show Answer & Explanation
Correct Answer: A
As x approaches 4, we substitute 4 into the function: f(4) = 2(4) + 3 = 8 + 3 = 11. Thus, the limit is 11.
Q4
Medium
What is the limit: lim(x→∞) (3x^3 - 2x)/(5x^3 + 4)?
A. 3/5
B. 1
C. 0
D. ∞
Show Answer & Explanation
Correct Answer: A
Divide every term by x^3. The limit then yields (3 - 0)/(5 + 0) = 3/5 as x approaches infinity.
Q5
Medium
Find the limit: lim(x→0) (1 - cos(x))/(x^2).
A. 0
B. 1/2
C. 1
D. Undefined
Show Answer & Explanation
Correct Answer: B
Using L'Hôpital's Rule or the identity 1 - cos(x) = 2sin^2(x/2), we find this limit equals 1/2.
Q6
Medium
Evaluate the limit: lim (x → 1) (x^2 - 1)/(x - 1).
A. 2
B. 1
C. 0
D. Undefined
Show Answer & Explanation
Correct Answer: A
By factoring the numerator, (x^2 - 1) becomes (x - 1)(x + 1). Cancelling (x - 1) gives us lim (x → 1) (x + 1) = 2.
Q7
Medium
Determine the limit: lim (x → 0) (sin(5x)/x).
A. 5
B. 1
C. 0
D. Undefined
Show Answer & Explanation
Correct Answer: A
Using the standard limit lim (x → 0) (sin(kx)/x) = k, where k = 5, the limit evaluates to 5.
Q8
Hard
Find the value of c that makes the function continuous at x = 1: f(x) = { x^2 - 1 if x < 1; cx + 2 if x >= 1. }
A. 1
B. 2
C. 3
D. 4
Show Answer & Explanation
Correct Answer: B
For f(x) to be continuous at x = 1, the left-hand limit (x^2 - 1 at x=1 gives 0) must equal the right-hand limit (cx + 2 at x=1 gives c + 2). Setting these equal gives c + 2 = 0, solving gives c = -2 which is option B.
Q9
Hard
Evaluate the limit: \( \lim_{x \to 0} \frac{\sin(5x)}{x} \).
A. 5
B. 0
C. 1
D. 25
Show Answer & Explanation
Correct Answer: A
Using the standard limit \( \lim_{x \to 0} \frac{\sin(kx)}{x} = k \) for any constant k, we substitute k = 5. Therefore, the limit evaluates to 5.
Q10
Hard
Find the value of \( L = \lim_{x \to 2} \frac{x^3 - 8}{x - 2} \).
A. 4
B. 8
C. 12
D. 16
Show Answer & Explanation
Correct Answer: C
The expression is indeterminate in the form \( \frac{0}{0} \). Factoring the numerator gives \( (x-2)(x^2 + 2x + 4) \), allowing us to cancel \( (x-2) \) and evaluate the limit directly, resulting in 12.

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Limits and Continuity — AP (Advanced Placement) AP Calculus BC Practice Questions Online

This page contains 136 practice MCQs for the chapter Limits and Continuity in AP (Advanced Placement) AP Calculus BC. The questions are organized by difficulty — 39 easy, 69 medium, 28 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.