Practice Exercise

A1.  is

(A) 1
(B) 0
(C) 
(D) ∞

A2. is

(A) 1
(B) 0
(C) –1
(D) ∞

A3.  is

(A) 0
(B) 1
(C) 
(D) ∞

A4.  is

(A) 1
(B) 0
(C) –1
(D) nonexistent

A5.  is

(A) 1
(B) 3
(C) 6
(D) ∞

A6.  is

(A) –2
(B) 
(C) 1
(D) 2

A7.  is

(A) –∞
(B) –1
(C) 3
(D) ∞ 

 A8.  is
(A) 0
(B) 1
(C) 3
(D) ∞ 

A9.  is

(A) –1
(B) 1
(C) 0
(D) ∞

A10.  is

(A) –1
(B) 1
(C) 0
(D) ∞

A11. 
(A) 
(B) = 1
(C) = 5
(D) does not exist

A12. 

(A) 
(B) = 1
(C) 
(D) does not exist

A13. 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 

A14. 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

A15.  is

(A) 1
(B) 
(C) 3
(D) ∞

A16.  is

(A) ∞
(B) 1
(C) –1
(D) nonexistent

A17. 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 line y = 2 is a horizontal asymptote.

A18.  is
(A) –2
(B) 1
(C) 2
(D) nonexistent

A19.  is
(A) 0
(B) 1
(C) –1
(D) nonexistent

A20.  is

(A) 0
(B) –1
(C) 1
(D) nonexistent

A21.  is

(A) 1
(B) 0
(C) π
(D) nonexistent

A22. 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 only
(D) I, II, and III

A23. If for x ≠ 0 and if f is continuous at x = 0, then k =

(A) –1
(B) 
(C) 
(D) 1

A24. Suppose  for x ≠ 1, 2

Then f(x) is continuous

(A) except at x = 1
(B) except at x = 2
(C) except at x = 1 or 2
(D) at each real number

A25. The graph of  has

(A) one vertical asymptote, at x = 1
(B) the y-axis as its vertical asymptote
(C) the x-axis as its horizontal asymptote and x = ±1 as its vertical asymptotes
(D) two vertical asymptotes, at x = ±1, but no horizontal asymptote

A26. 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  and two vertical asymptotes, at x = ±1

A27. Let 

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) I, II, and III

B1. If [x] denotes the greatest integer not greater than x, then  is

(A) 
(B) 1
(C) 0
(D) nonexistent

B2. If [x] denotes the greatest integer not greater than x, then  is

(A) –3
(B) –2
(C) –1
(D) nonexistent

B3. 

(A) is –1
(B) is infinity
(C) is zero
(D) does not exist

B4. 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

Questions B5–B9 are based on the function f shown in the graph and defined below:

B5.  

(A) equals 0
(B) equals 1
(C) equals 2
(D) does not exist

B6.The function f is defined on [–1,3]

(A) if x ≠ 0
(B) if x ≠ 1
(C) if x ≠ 2
(D) at each x in [–1,3]

B7. The function f has a removable discontinuity at

(A) x = 0
(B) x = 1
(C) x = 2
(D) x = 3

B8. 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

B9. The function f has a jump discontinuity at

(A) x = –1
(B) x = 1
(C) x = 2
(D) x = 3

B10. 

CHALLENGE B11. 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) I only
(B) III only
(C) I and III only
(D) I, II, and III

CHALLENGE B12. If , then 

(A) = 0
(B) 
(C) 
(D) does not exist

Questions B13–B15 are based on the function f shown in the graph.

B13. 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) 2 only
(C) –1 or 1 only
(D) –1 or 2 only

B14.  does not exist for a =

(A) –1 only
(B) 1 only
(C) 2 only
(D) 1 and 2 only

B15. Which of the following statements about limits at x = 1 is (are) true?

(A) I only
(B) II only
(C) I and II only
(D) I, II, and III