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