## Calculus AB and Calculus BC

## CHAPTER 2 Limits and Continuity

### Practice Exercises

**Part A. Directions:** Answer these questions *without* using your calculator.

**1.**

**(A)** 1

**(B)** 0

**(C)**

**(D)** −1

**(E)** ∞

**2.**

**(A)** 1

**(B)** 0

**(C)** −4

**(D)** −1

**(E)** ∞

**3.**

**(A)** 0

**(B)** 1

**(C)**

**(D)** ∞

**(E)** none of these

**4.**

**(A)** 1

**(B)** 0

**(C)** ∞

**(D)** −1

**(E)** nonexistent

**5.**

**(A)** 4

**(B)** 0

**(C)** 1

**(D)** 3

**(E)** ∞

**6.**

**(A)** −2

**(B)**

**(C)** 1

**(D)** 2

**(E)** nonexistent

**7.**

**(A)** −∞

**(B)** −1

**(C)** 0

**(D)** 3

**(E)** ∞

**8.**

**(A)** 3

**(B)** ∞

**(C)** 1

**(D)** −1

**(E)** 0

**9.**

**(A)** −1

**(B)** 1

**(C)** 0

**(D)** ∞

**(E)** none of these

**10.**

**(A)** −1

**(B)** 1

**(C)** 0

**(D)** ∞

**(E)** none of these

**11.**

**(A)** = 0

**(B)**

**(C)** = 1

**(D)** = 5

**(E)** does not exist

**12.**

**(A)** = 0

**(B)**

**(C)** = 1

**(D)**

**(E)** does not exist

**13.** The graph of *y* = arctan *x* has

**(A)** vertical asymptotes at *x* = 0 and *x* = π

**(B)** horizontal asymptotes at

**(C)** horizontal asymptotes at *y* = 0 and *y* = π

**(D)** vertical asymptotes at

**(E)** none of these

**14.** The graph of has

**(A)** a vertical asymptote at *x* = 3

**(B)** a horizontal asymptote at

**(C)** a removable discontinuity at *x* = 3

**(D)** an infinite discontinuity at *x* = 3

**(E)** none of these

**15.**

**(A)** 1

**(B)**

**(C)** 3

**(D)** ∞

**(E)**

**16.**

**(A)** ∞

**(B)** 1

**(C)** nonexistent

**(D)** −1

**(E)** none of these

**17.** Which statement is true about the curve

**(A)** The line is a vertical asymptote.

**(B)** The line *x* = 1 is a vertical asymptote.

**(C)** The line is a horizontal asymptote.

**(D)** The graph has no vertical or horizontal asymptote.

**(E)** The line *y* = 2 is a horizontal asymptote.

**18.**

**(A)** −4

**(B)** −2

**(C)** 1

**(D)** 2

**(E)** nonexistent

**19.**

**(A)** 0

**(B)** nonexistent

**(C)** 1

**(D)** −1

**(E)** none of these

**20.**

**(A)** 0

**(B)** ∞

**(C)** nonexistent

**(D)** −1

**(E)** 1

**21.**

**(A)** 1

**(B)** 0

**(C)** ∞

**(D)** nonexistent

**(E)** none of these

**22.** Let

Which of the following statements is (are) true?

I. exists

II. *f* (1) exists

III. *f* is continuous at *x* = 1

**(A)** I only

**(B)** II only

**(C)** I and II

**(D)** none of them

**(E)** all of them

**23.**

and if *f* is continuous at *x* = 0, then *k* =

**(A)** −1

**(B)**

**(C)** 0

**(D)**

**(E)** 1

**24.**

Then *f* (*x*) is continuous

**(A)** except at *x* = 1

**(B)** except at *x* = 2

**(C)** except at *x* = 1 or 2

**(D)** except at *x* = 0, 1, or 2

**(E)** at each real number

**25.** The graph of has

**(A)** one vertical asymptote, at *x* = 1

**(B)** the *y*-axis as vertical asymptote

**(C)** the *x*-axis as horizontal asymptote and *x* = ±1 as vertical asymptotes

**(D)** two vertical asymptotes, at *x* = ±1, but no horizontal asymptote

**(E)** no asymptote

**26.** The graph of has

**(A)** a horizontal asymptote at but no vertical asymptote

**(B)** no horizontal asymptote but two vertical asymptotes, at *x* = 0 and *x* = 1

**(C)** a horizontal asymptote at and two vertical asymptotes, at *x* = 0 and *x* = 1

**(D)** a horizontal asymptote at *x* = 2 but no vertical asymptote

**(E)** a horizontal asymptote at and two vertical asymptotes, at *x* = ±1

**27.**

Which of the following statements is (are) true?

I. *f* (0) exists

II. exists

III. *f* is continuous at *x* = 0

**(A)** I only

**(B)** II only

**(C)** I and II only

**(D)** all of them

**(E)** none of them

**Part B. Directions:** Some of the following questions require the use of a graphing calculator.

**28.** If [*x*] is the greatest integer not greater than *x*, then is

**(A)**

**(B)** 1

**(C)** nonexistent

**(D)** 0

**(E)** none of these

**29.** (With the same notation) is

**(A)** −3

**(B)** −2

**(C)** −1

**(D)** 0

**(E)** none of these

**30.**

**(A)** is −1

**(B)** is infinity

**(C)** oscillates between −1 and 1

**(D)** is zero

**(E)** does not exist

**31.** The function

**(A)** is continuous everywhere

**(B)** is continuous except at *x* = 0

**(C)** has a removable discontinuity at *x* = 0

**(D)** has an infinite discontinuity at *x* = 0

**(E)** has *x* = 0 as a vertical asymptote

Questions 32–36 are based on the function *f* shown in the graph and defined below:

**32.**

**(A)** equals 0

**(B)** equals 1

**(C)** equals 2

**(D)** does not exist

**(E)** none of these

**33.** The function *f* is defined on [−1,3]

**(A)** if *x* ≠ 0

**(B)** if *x* ≠ 1

**(C)** if *x* ≠ 2

**(D)** if *x* ≠ 3

**(E)** at each *x* in [−1,3]

**34.** The function *f* has a removable discontinuity at

**(A)** *x* = 0

**(B)** *x* = 1

**(C)** *x* = 2

**(D)** *x* = 3

**(E)** none of these

**35.** On which of the following intervals is *f* continuous?

**(A)** −1 ≤ *x* ≤ 0

**(B)** 0 < *x* < 1

**(C)** 1 ≤ *x* ≤ 2

**(D)** 2 ≤ *x* ≤ 3

**(E)** none of these

**36.** The function *f* has a jump discontinuity at

**(A)** *x* = −1

**(B)** *x* = 1

**(C)** *x* = 2

**(D)** *x* = 3

**(E)** none of these

**CHALLENGE**

**37.**

**(A)** −∞

**(B)**

**(C)**

**(D)** ∞

**(E)** none of these

**38.** Suppose and *f* (−3) is not defined. Which of the following statements is (are) true?

I.

II. *f* is continuous everywhere except at *x* = −3.

III. *f* has a removable discontinuity at *x* = −3.

**(A)** None of them

**(B)** I only

**(C)** III only

**(D)** I and III only

**(E)** All of them

**CHALLENGE**

**39.** If then *y* is

**(A)** 0

**(B)**

**(C)**

**(D)**

**(E)** nonexistent

Questions 40–42 are based on the function *f* shown in the graph.

**40.** For what value(s) of *a* is it true that exists and *f* (*a*) exists, but It is possible that *a* =

**(A)** −1 only

**(B)** 1 only

**(C)** 2 only

**(D)** −1 or 1 only

**(E)** −1 or 2 only

**41.** does not exist for *a* =

**(A)** −1 only

**(B)** 1 only

**(C)** 2 only

**(D)** 1 and 2 only

**(E)** −1, 1, and 2

**42.** Which statements about limits at *x* = 1 are true?

I. exists.

II. exists.

III. exists.

**(A)** none of I, II, or III

**(B)** I only

**(C)** II only

**(D)** I and II only

**(E)** I, II, and III

### Practice Exercises

**Part A. Directions:** Answer these questions *without* using your calculator.

**1.**

**(A)** 1

**(B)** 0

**(C)**

**(D)** −1

**(E)** ∞

**2.**

**(A)** 1

**(B)** 0

**(C)** −4

**(D)** −1

**(E)** ∞

**3.**

**(A)** 0

**(B)** 1

**(C)**

**(D)** ∞

**(E)** none of these

**4.**

**(A)** 1

**(B)** 0

**(C)** ∞

**(D)** −1

**(E)** nonexistent

**5.**

**(A)** 4

**(B)** 0

**(C)** 1

**(D)** 3

**(E)** ∞

**6.**

**(A)** −2

**(B)**

**(C)** 1

**(D)** 2

**(E)** nonexistent

**7.**

**(A)** −∞

**(B)** −1

**(C)** 0

**(D)** 3

**(E)** ∞

**8.**

**(A)** 3

**(B)** ∞

**(C)** 1

**(D)** −1

**(E)** 0

**9.**

**(A)** −1

**(B)** 1

**(C)** 0

**(D)** ∞

**(E)** none of these

**10.**

**(A)** −1

**(B)** 1

**(C)** 0

**(D)** ∞

**(E)** none of these

**11.**

**(A)** = 0

**(B)**

**(C)** = 1

**(D)** = 5

**(E)** does not exist

**12.**

**(A)** = 0

**(B)**

**(C)** = 1

**(D)**

**(E)** does not exist

**13.** The graph of *y* = arctan *x* has

**(A)** vertical asymptotes at *x* = 0 and *x* = π

**(B)** horizontal asymptotes at

**(C)** horizontal asymptotes at *y* = 0 and *y* = π

**(D)** vertical asymptotes at

**(E)** none of these

**14.** The graph of has

**(A)** a vertical asymptote at *x* = 3

**(B)** a horizontal asymptote at

**(C)** a removable discontinuity at *x* = 3

**(D)** an infinite discontinuity at *x* = 3

**(E)** none of these

**15.**

**(A)** 1

**(B)**

**(C)** 3

**(D)** ∞

**(E)**

**16.**

**(A)** ∞

**(B)** 1

**(C)** nonexistent

**(D)** −1

**(E)** none of these

**17.** Which statement is true about the curve

**(A)** The line is a vertical asymptote.

**(B)** The line *x* = 1 is a vertical asymptote.

**(C)** The line is a horizontal asymptote.

**(D)** The graph has no vertical or horizontal asymptote.

**(E)** The line *y* = 2 is a horizontal asymptote.

**18.**

**(A)** −4

**(B)** −2

**(C)** 1

**(D)** 2

**(E)** nonexistent

**19.**

**(A)** 0

**(B)** nonexistent

**(C)** 1

**(D)** −1

**(E)** none of these

**20.**

**(A)** 0

**(B)** ∞

**(C)** nonexistent

**(D)** −1

**(E)** 1

**21.**

**(A)** 1

**(B)** 0

**(C)** ∞

**(D)** nonexistent

**(E)** none of these

**22.** Let

Which of the following statements is (are) true?

I. exists

II. *f* (1) exists

III. *f* is continuous at *x* = 1

**(A)** I only

**(B)** II only

**(C)** I and II

**(D)** none of them

**(E)** all of them

**23.**

and if *f* is continuous at *x* = 0, then *k* =

**(A)** −1

**(B)**

**(C)** 0

**(D)**

**(E)** 1

**24.**

Then *f* (*x*) is continuous

**(A)** except at *x* = 1

**(B)** except at *x* = 2

**(C)** except at *x* = 1 or 2

**(D)** except at *x* = 0, 1, or 2

**(E)** at each real number

**25.** The graph of has

**(A)** one vertical asymptote, at *x* = 1

**(B)** the *y*-axis as vertical asymptote

**(C)** the *x*-axis as horizontal asymptote and *x* = ±1 as vertical asymptotes

**(D)** two vertical asymptotes, at *x* = ±1, but no horizontal asymptote

**(E)** no asymptote

**26.** The graph of has

**(A)** a horizontal asymptote at but no vertical asymptote

**(B)** no horizontal asymptote but two vertical asymptotes, at *x* = 0 and *x* = 1

**(C)** a horizontal asymptote at and two vertical asymptotes, at *x* = 0 and *x* = 1

**(D)** a horizontal asymptote at *x* = 2 but no vertical asymptote

**(E)** a horizontal asymptote at and two vertical asymptotes, at *x* = ±1

**27.**

Which of the following statements is (are) true?

I. *f* (0) exists

II. exists

III. *f* is continuous at *x* = 0

**(A)** I only

**(B)** II only

**(C)** I and II only

**(D)** all of them

**(E)** none of them

**Part B. Directions:** Some of the following questions require the use of a graphing calculator.

**28.** If [*x*] is the greatest integer not greater than *x*, then is

**(A)**

**(B)** 1

**(C)** nonexistent

**(D)** 0

**(E)** none of these

**29.** (With the same notation) is

**(A)** −3

**(B)** −2

**(C)** −1

**(D)** 0

**(E)** none of these

**30.**

**(A)** is −1

**(B)** is infinity

**(C)** oscillates between −1 and 1

**(D)** is zero

**(E)** does not exist

**31.** The function

**(A)** is continuous everywhere

**(B)** is continuous except at *x* = 0

**(C)** has a removable discontinuity at *x* = 0

**(D)** has an infinite discontinuity at *x* = 0

**(E)** has *x* = 0 as a vertical asymptote

Questions 32–36 are based on the function *f* shown in the graph and defined below:

**32.**

**(A)** equals 0

**(B)** equals 1

**(C)** equals 2

**(D)** does not exist

**(E)** none of these

**33.** The function *f* is defined on [−1,3]

**(A)** if *x* ≠ 0

**(B)** if *x* ≠ 1

**(C)** if *x* ≠ 2

**(D)** if *x* ≠ 3

**(E)** at each *x* in [−1,3]

**34.** The function *f* has a removable discontinuity at

**(A)** *x* = 0

**(B)** *x* = 1

**(C)** *x* = 2

**(D)** *x* = 3

**(E)** none of these

**35.** On which of the following intervals is *f* continuous?

**(A)** −1 ≤ *x* ≤ 0

**(B)** 0 < *x* < 1

**(C)** 1 ≤ *x* ≤ 2

**(D)** 2 ≤ *x* ≤ 3

**(E)** none of these

**36.** The function *f* has a jump discontinuity at

**(A)** *x* = −1

**(B)** *x* = 1

**(C)** *x* = 2

**(D)** *x* = 3

**(E)** none of these

**CHALLENGE**

**37.**

**(A)** −∞

**(B)**

**(C)**

**(D)** ∞

**(E)** none of these

**38.** Suppose and *f* (−3) is not defined. Which of the following statements is (are) true?

I.

II. *f* is continuous everywhere except at *x* = −3.

III. *f* has a removable discontinuity at *x* = −3.

**(A)** None of them

**(B)** I only

**(C)** III only

**(D)** I and III only

**(E)** All of them

**CHALLENGE**

**39.** If then *y* is

**(A)** 0

**(B)**

**(C)**

**(D)**

**(E)** nonexistent

Questions 40–42 are based on the function *f* shown in the graph.

**40.** For what value(s) of *a* is it true that exists and *f* (*a*) exists, but It is possible that *a* =

**(A)** −1 only

**(B)** 1 only

**(C)** 2 only

**(D)** −1 or 1 only

**(E)** −1 or 2 only

**41.** does not exist for *a* =

**(A)** −1 only

**(B)** 1 only

**(C)** 2 only

**(D)** 1 and 2 only

**(E)** −1, 1, and 2

**42.** Which statements about limits at *x* = 1 are true?

I. exists.

II. exists.

III. exists.

**(A)** none of I, II, or III

**(B)** I only

**(C)** II only

**(D)** I and II only

**(E)** I, II, and III