Timedora
Free Practice Quiz Question List

2 Limits and Continuity Online Quiz Questions

Use this free practice quiz with 20 questions to review 2 Limits and Continuity, test your knowledge, and prepare for your next test or exam.

20 questions
01
Choose one
1 point

Which statement best explains what lim⁡x→af(x)=L\lim_{x\to a}f(x)=L means?

  1. A

    The limit must equal the value of the function at the target.

  2. B

    The limit concerns nearby function values and may exist even when the function is undefined at the target.

  3. C

    A limit exists only when the function is defined at the target.

  4. D

    The limit is the slope of the graph at the target.

02
Choose one
1 point

Evaluate lim⁡x→∞2x2+1x2−3\lim_{x\to\infty}\frac{2x^2+1}{x^2-3}.

  1. A

    00

  2. B

    11

  3. C

    22

  4. D

    The limit does not exist because both degrees are equal.

03
Choose one
1 point

Which set of conditions is required for a function to be continuous at x=ax=a?

  1. A

    Only f(a)f(a) must be defined.

  2. B

    The left- and right-hand limits may differ as long as f(a)f(a) is defined.

  3. C

    The limit must exist, but it need not equal f(a)f(a).

  4. D

    All three conditions must hold: f(a)f(a) is defined, the limit exists, and the limit equals f(a)f(a).

04
Choose all
1 point

Select all conditions that are sufficient for the two-sided limit lim⁡x→af(x)\lim_{x\to a}f(x) to exist.

  1. A

    The left-hand and right-hand limits are equal.

  2. B

    The function must be defined at aa.

  3. C

    Both one-sided limits exist.

  4. D

    The graph must contain the point (a,L)(a,L).

05
Choose all
1 point

Select all algebraic techniques identified in the material as useful for evaluating indeterminate limits of suitable expressions.

  1. A

    Factoring and canceling common factors can help.

  2. B

    Substituting the target value repeatedly always gives an exact result.

  3. C

    Multiplying every expression by its denominator is a valid general method.

  4. D

    Multiplying by a conjugate can help with expressions involving radicals.

06
True or false
1 point

A function can have a removable discontinuity at x=ax=a even if its two-sided limit exists.

  1. A

    True

  2. B

    False

07
True or false
1 point

Because lim⁡x→0+1x=∞\lim_{x\to 0^+}\frac{1}{x}=\infty and lim⁡x→0−1x=−∞\lim_{x\to 0^-}\frac{1}{x}=-\infty, the two-sided limit lim⁡x→01x\lim_{x\to 0}\frac{1}{x} exists as an infinite limit.

  1. A

    True

  2. B

    False

08
Written response
1 point

Find the exact value of lim⁡x→2x2−4x−2\lim_{x\to 2}\frac{x^2-4}{x-2}.

09
Fill in the blank
1 point

A graph approaches y=3y=3 from both sides as x→4x\to4, but the point at x=4x=4 is missing. The limit is lim⁡x→4f(x)=\lim_{x\to4}f(x)=.

10
Fill in the blank
1 point

For f(x)={x2,x<12x−1,x≥1f(x)=\begin{cases}x^2,&x<1\\2x-1,&x\ge1\end{cases}, the function is at x=1x=1.

11
Open ended
1 point

Use the Intermediate Value Property to explain why the function f(x)=x3−x−1f(x)=x^3-x-1 has at least one zero in the interval (1,2)(1,2).

12
Written response
1 point

Find the exact value of lim⁡x→3(2x2−x+4)\lim_{x\to3}(2x^2-x+4).

13
Choose one
1 point

For f(x)=x2−1x−1f(x)=\frac{x^2-1}{x-1}, which statement correctly describes the behavior as x→1x\to1?

  1. A

    The limit does not exist because the original function is undefined at x=1x=1.

  2. B

    The limit is 22, and the graph has a removable discontinuity at (1,2)(1,2).

  3. C

    The limit is 11, because the denominator approaches zero.

  4. D

    The limit is infinite, and x=1x=1 is a vertical asymptote.

14
Choose one
1 point

Evaluate lim⁡x→3(2x2−x+4)\displaystyle\lim_{x\to 3}(2x^2-x+4).

  1. A

    17

  2. B

    19

  3. C

    21

  4. D

    25

15
True or false
1 point

True or false: A two-sided limit can exist even if the function is undefined at the value being approached.

  1. A

    True

  2. B

    False

16
Choose one
1 point

Evaluate lim⁡x→2x2−4x−2\displaystyle\lim_{x\to 2}\frac{x^2-4}{x-2}.

  1. A

    0

  2. B

    2

  3. C

    4

  4. D

    Does not exist

17
Choose one
1 point

For f(x)={x+1,x<2,5−x,x≥2,\displaystyle f(x)=\begin{cases}x+1,&x<2,\\5-x,&x\ge2, \end{cases} evaluate lim⁡x→2f(x)\displaystyle\lim_{x\to2}f(x).

  1. A

    2

  2. B

    3

  3. C

    4

  4. D

    The limit does not exist

18
Choose one
1 point

Evaluate lim⁡x→0x+1−1x\displaystyle\lim_{x\to0}\frac{\sqrt{x+1}-1}{x}.

  1. A

    12\frac{1}{2}

  2. B

    11

  3. C

    22

  4. D

    00

19
Written response
1 point

What type of discontinuity does f(x)=1x−1f(x)=\frac{1}{x-1} have at x=1x=1?

20
Written response
1 point

Find the exact value of lim⁡x→01+2x−1x\displaystyle\lim_{x\to0}\frac{\sqrt{1+2x}-1}{x}. Enter the answer as a number; the absolute tolerance is 00.