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Free Practice Quiz Question List

Limits and Continuity Online Quiz Questions

Use this free practice quiz with 20 questions to review Limits and Continuity, test your knowledge, and prepare for your next test or exam.

20 questions
01
True or false
1 point

True or false: The value of f(a) must equal L for the limit of f(x) as x approaches a to equal L.

  1. A

    True

  2. B

    False

02
True or false
1 point

True or false: If direct substitution into a limit produces 0/0, then the value of the limit is 0.

  1. A

    True

  2. B

    False

03
Choose one
1 point

What is the value of the limit as x approaches 2 of 3x^2 - 5x + 1?

  1. A

    1

  2. B

    2

  3. C

    3

  4. D

    4

04
Written response
1 point

Find the exact value of lim x→3 (x² − 9)/(x − 3). Enter the value as a decimal or integer; no tolerance is allowed.

05
Choose one
1 point

For f(x) = 2x + 1 when x < 3 and f(x) = 7 − x when x ≥ 3, what can be concluded about lim x→3 f(x)?

  1. A

    The limit is 4

  2. B

    The two-sided limit does not exist

  3. C

    The limit is 7

  4. D

    The function is continuous at x = 3

06
Choose all
1 point

Select all conditions that are required for a function f to be continuous at x = a.

  1. A

    f(a) is defined

  2. B

    lim x→a f(x) exists

  3. C

    f has a derivative at x = a

  4. D

    lim x→a f(x) = f(a)

07
Fill in the blank
1 point

A hole in a graph, where the finite two-sided limit exists but the function is at the point, is called a .

08
Choose one
1 point

For f(x) = (x + 1)/[(x − 2)(x + 3)], which statement correctly describes the behavior near x = 2?

  1. A

    −∞ from the left and +∞ from the right

  2. B

    +∞ from the left and −∞ from the right

  3. C

    −∞ from the left and +∞ from the right, with x = 2 as a vertical asymptote

  4. D

    The limit is 0 from both sides

09
Written response
1 point

Find the exact value of lim x→∞ (5x³ − 2x + 1)/(2x³ + 7x² − 4). Enter the value as a decimal or integer; no tolerance is allowed.

10
Choose all
1 point

Select all statements that correctly describe the Intermediate Value Theorem.

  1. A

    The function is continuous on the entire closed interval

  2. B

    The function must be differentiable on the interval

  3. C

    The target value lies between the endpoint values

  4. D

    The theorem guarantees at least one point c in the interval with f(c) equal to the target

11
Fill in the blank
1 point

In the formal definition of lim x→a f(x) = L, measures how close f(x) must be to L, while measures how close x must be to a.

12
Open ended
1 point

Let f(x) = x² + 1 for x < 2 and f(x) = ax + 3 for x ≥ 2. Determine the value of a that makes f continuous at x = 2, and justify your answer using the continuity conditions.

13
Choose one
1 point

Which condition correctly characterizes continuity of a function on the closed interval [a,b]?

  1. A

    The function must have equal two-sided limits at both endpoints

  2. B

    The function must be continuous on (a,b), right-continuous at a, and left-continuous at b

  3. C

    The function must be differentiable at every point of [a,b]

  4. D

    The function must have a horizontal asymptote

14
Choose one
1 point

Suppose that lim as x approaches 2 from the left of f(x) equals 4 and lim as x approaches 2 from the right of f(x) also equals 4, while f(2) is undefined. What is the two-sided limit as x approaches 2 of f(x)?

  1. A

    The limit does not exist because f(2) is undefined.

  2. B

    The limit exists and equals 4.

  3. C

    The limit exists and equals 7.

  4. D

    The limit must equal f(2).

15
Choose one
1 point

Let f(x) = x² + 1 for x < 2 and f(x) = ax + 3 for x ≥ 2. Which value of a makes f continuous at x = 2?

  1. A

    -1

  2. B

    0

  3. C

    1

  4. D

    4

16
Choose one
1 point

Evaluate the limit as x approaches positive infinity of (5x³ − 2x + 1) divided by (2x³ + 7x² − 4).

  1. A

    5/2

  2. B

    2/5

  3. C

    0

  4. D

    There is no finite limit

17
Choose one
1 point

For f(x) = x³ + x − 1 on [0, 1], what conclusion is justified by the Intermediate Value Theorem?

  1. A

    The function is discontinuous because its values change sign.

  2. B

    The IVT proves that there is exactly one root in [0, 1].

  3. C

    The IVT applies only if the endpoint values are equal.

  4. D

    The IVT proves that at least one root lies in (0, 1).

18
True or false
1 point

True or false: The statement that the limit of 1 divided by (x - 2) as x approaches 2 from the right is positive infinity describes a limit at infinity.

  1. A

    True

  2. B

    False

19
Written response
1 point

Evaluate the limit as x approaches 3 of (x² − 9) divided by (x − 3). Enter the exact numerical value as a decimal or integer; no tolerance is allowed.

20
Written response
1 point

What standard term describes a discontinuity where the finite two-sided limit exists but the function is undefined at the point or has a different assigned value there?