·102 ;;:..p CalculusI •

Part A. Directions: Answer these questions without using your calculator.x2 - 41. lirn2""4" is

x-->2 X +

1(A) 1 (B) 0 (C) -,- (D) -1 (E).00

2

2. 4 -r X2lirn-2-1 isX---7(X1 X -

(A) 1 (B) 0 (C) -4 (D) -1 (E) 00

P x-33. lirn 2 2 3 is-4 .<-->3 X - x-

1'I I (A) 0 (B) 1 (C) (D) (E) none of these'::? - 00

4 4

J 4. lim -i- isx-->o

(A) 1 (B) 0 (C) 00 (D) -1 (E) nonexistent

x3 - 85. lirn~ isx-->2 x -

Ai l (A) 4 (B) 0 (C) 1 (D) 3 (E) 00

~

d ,,- 4 - X2'7. 6. lirn 2 is() x,-->~ 4x - x-2

~

(A) -2 (B) 1 (E)J}-- (C) 1 (D) 2 nonexistent

4

.~ 5x3+ 277. lirn is

j x-->~ 20x2 + lOx +9

(A) -00 (B) -1 (C) 0 (D) 3 (E) 00

3x2+ 278. lirn 3 27 isx-4oo x-

(A) 3 (B) 00 (C) 1 (D) -1 (E) 0

Practice Exercises

9. 2-xlirn~ isx~oo _

(A) -1 (B) 1

10. 2-xlirn2 isx-+--oo x

(A) -1 (B) 1

(C) 0 (D) 00 (E) none of these

(C) 0 (D) 00 (E) none of these

•Limits and Continuity 103

, 11. lim sin5x,...•0 x• (A) =0 (B) 1

5

12. lim sin2x, -e0 3x

(A) =0 (B) 23

(C) (D)=1 =5 (E) does not exist

3- 2(C) =1 (D) (E) does not exist

13. The graph of y = arctan x has

(A) vertical asymptotes at x = 0 and x = 1t

(B) horizontal asymptotes at y = ±~(C) horizontal asymptotes at y = 0 and y = 1t

(D) vertical asymptotes atx = ±~(E) none of these

x2-9The graph of y = -- has

3x-9

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

(C) a removable discontinuity at x = 3(E) none of these

a horizontal asymptote at y = !3

(D) an infinite discontinuity at x = 3

'5. l' sinx ,Im-2-3- ISx-->o X + X

(A) 1 (B)1-3

16. li . 1 .m sm x ISX-->O

(A) 00 (B) 1

(C) 3 (E)1

4(D) 00

(C) nonexistent (D) -1 (E) none of these

17. Which statement is true about the curve y = 2 :;: : :x2 ?

(A) The line x = - i is a vertical asytnptote.(B) The line x = 1 is a vertical asymptote.(C) The line y = -i is a horizontal asymptote.(D) The graph has no vertical or horizontal asymptote.(E) The line y = 2 is a horizontal asymptote.

18. 1. 2x2+1im is

x-->~ (2 -x)(2 +x)

(A) -4 (B) -2 (C) 1 (D) 2 (E) nonexistent

c

104 AP Calculus

19. r Ixi.un- lSx~o x

(A) ° (B) nonexistent (C) 1 (D) -1

20. limxsin± isx--;~

(A) ° (B) 00 (C) nonexistent (D) -1

21. limsin(1t-x)

isX--;7t 7t-x

(E) none of these

(E) 1

(A) 1 (C) 00 (E) none of these(B) ° (D) nonexistent

22. Let f(x) = {~~11 ifx~14 ifx=l.

Which of the following statements is (are) true?

I. limf(x) existsx--;i

II. f(l) existsIII. f is continuous at x = 1

(A) I only (B)(D) none of them

II only (C) I and II(E) all of them

23. If {f(X) : x~:xfor x :;to,f(O) - k,

and iff is continuous at x = 0, then k =

(A) -112

(D) 12

(E) 1(B) (C) °

jf(X) = x~x~~.~22for X oF 1,2,

24. Suppose f(1) = -3,

f(2) = 4.

Thenf(x) is continuous

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

except at x = 2 (C) except at x = 1 or 2(E) at each real number

425. The graph of f(x) = X2 -1 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

Limits and Continuity 105

2x2+2x+326. The graph of y = 4xL 4x has

• (A) a horizontal asymptote at y = + ~ but no vertical asymptote(B) no horizontal asymptote but two vertical asymptotes, at x = 0 and x = 1

(C) a horizontal asymptote at y = ~ 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 Y = ~ and two vertical asymptotes, at x = ±1

fart B. Directions: Some of the following questions require the use of a graphingralCUlator._8. If [x] is the greatest integer not greater than x, then lim [x] is

• x ...•lf2

(A) 12

27 L f(X)={X2;X ifx*O.. . et1 ifx=O

Which of the following statements is (are) true?

1. f(O) exists

II. limf(x) existsx ...•o

III. f is continuous at x = 0

(A)(D)

I only (B)all of them

II only (C) I and II only(E) none of them

(B) (C) nonexistent1 (D) 0 (E) none of these

29. (With the same notation), lijn].r] isx ...•-2

, i-

(A) -3 (B) -2 (C) -1 (D) 0 (E) none of these

30. limsinx

(A) is-l(D) is zero

(B) is infinity (C) oscillates between -1 and 1(E) does not exist

31. The functionf(x) = {X2 / x (x * 0)o (x = 0)

(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

s

1,06 AP CalculusII Questions 32-36 are based on the

function f shown in the graph anddefined below:

y

I-x . (-1::::; x < 0)2X2_2 (O::::; x::::;1)

f(x) = -x+2 (1 < x < 2)1 (x = 2) x

2x-4 (2 < x::::;3) -1 0

32. limf(x)x ...•2

(A) equals 0 (B) equals 1 (C) equals 2(D) does not exist (E) none of these

33. The functionfis defined on [-1,3]

(A) if x;t: 0(D) if x;t: 3

(B) if x;t: 1 (C) if x;t: 2(E) at each x in [-1,3]

34. The functionfhas 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(D) 2::::;x::::; 3

(B) 0 < x < 1 (C) 1::::;x ::::;2(E) none of these

36. The functionfhas a jump discontinuity at

(A) x=-l(D) x = 3

(B) x = 1 (C) x = 2(E) none of these