Set 2: Multiple-Choice Questions on Limits and Continuity

I' ~ - 4, 1. Im~4IS .-2 x- + 4

(A) 1 (8) 0

2. lim ; - .t12 is ~.... -

(e) _1 2

(D) -1 (E) 00

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

3 r z-3 . • .~ ~ _ 2.t - 3 IS

(A) 0 (B) 1 (e) 1 (D) 00 (E) none of these 4

4. lim ~ is .-0 z

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

5 r ,.3-8, • .!!1l T-4 15

(A) 4 (B) 0 (e) 1 (D) 3 (E) 00

L. I' 4 - ,xl , u. 1m ,_.2 IS .-- -.z- - .t - 2

(A) - 2 (8) - 1 (e) 1 (D) 2 4

(E) nonexistent

7 r Sz'+27 , • .!,g! 20? + J(a + 9 IS

(A) 00 (8) 1 (C) 3 (D) 0 (E) 1 4

8 I, 3z2 + 27 ' . .~ r - 27 15

(A) 3 (8) 00 (C) 1 (D) - 1 (E) 0

2-' 9. lim -2' is ,--

(A) - I (B) 1 (e) 0 (D) 00 (E) none of these

· . 10. If [x] is the greatest integer not greater than x, then lim [x] is

.I-a,l

(A) 1 (B) 1 (e) noneltistent (D) 0 (E) none of these 2

11. (With the same notation) lim [x] is .-2 (A) 0 (B) 1 (e) 2 (D) 3 (E) none of these

12. lim !!!!..3 is .-0 x

(A) 0 (D) (e) 'f/' CD) 00 (E) The limit does not eltist.

13. r sin 2x . 1m --IS .-0 x

(A) (D) 2 (e) 1 (D) 0 (E) 01:)

2

14. lim sinx .1' .... -

(A) is nonexistent (8) is infinity (C) oscillates between -1 and 1

(D) is zero (E) is 1 or-l

IS. I" sin l.t . un ~ IS .-0 Sin x

(A) (D) 4 (e) 3 (D) 0 (E) nonexistent 3 4

16. r I - cos x . 1m IS .-0 x

(A) nonexistent (B) 1 (e) 2 (D) 00 (E) none of these

17. I" sin x . Im--1S .-0 xl + 3x

(A) (D) 1 (C) 3 (D) 00 (E) ! 3 4

18. lim sin 1 is .-0 x

(A) 00 (8) (e) nonexistent (D) -I (E) none of these

19. r tan 'lrX • Im--IS .-0 x

(A) .! 11'

(8) 0 (e) 1 (D) 'f/' (E) 00

10. r 2· 1 1m x SlD-.1' .... - X

(A) is 1 (B) is 0 (C) is 00 (D) oscillates between - 1 and 1 (E) is none of these

21. lim xcscx is

22.

23.

24.

25.

,,_0 (A) - co (B) - 1 (e) 0 (D) 1 (E) co

lim 2X2 + 1 is x_- (2-x)(2+x)

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

lim.B is x_o x

(A) 0 (B) nonexistent (e) 1 (D) - 1 (E) none of these

1. . 1 . 1m XSIn- IS x_.. X

(A) 0 (B) 00

1. sin(7t-x). 1m IS

x .... " 7t-X (A) 1 (B) 0

(e) nonexistent (D) - 1 (E) 1

(e) 00 (D) nonexistent (E) none of these

26. If [xl is the greatest integer in x, then what is lim [x + I]? x .... -I

(A) -1 (B) 0 (e) 1 (D) 2 (E) The limit does not exist.

27. Let lex) = {X2 -1 if x -:1: 1. 4 ifx=l.

Which of the following statements. I, II, and III, are true? I. lim/ex) exists II. f(l) exists III.fis continuous at x = 1

x_I

(A) only I (B) only II (e) I and II (D) none of them (E) all of them

{/(X) == x2-x forx~O,

29. If 2x f{O) = k,

and if I is continuous at x == 0, then k ==

1 1 (A) -1 (B) 2 (e) 0 (D) 2 (E) 1

3x(x-l) /(x) = 2 for x -:1: 1, 2.

x -3x+2 JO. Suppose l(l) = - 3,

/(2) = 4.

Then lex) is continuous

(A) except at x = 1 (8) except at x ;:; 2 (e) except at x == 1 or 2 (D) except at x = 0, 1, or 2 (E) at each real number

Answers for Set 2: Limits and Continuity 1. B 6. B 11. E 16. E 21. D 26. E 2. D 7. A 12. B 17. B 22. B 27. C 3. C 8. E 13. B 18. C 23. A 28. A 4. A 9. C 14. A 19. D 24. E 29. B 5. 0 10. 0 15. C 20. C 25. A 30. B

Set 3: Multiple-Choice Questions on Differentiation

In each of Questions 1-27 a function is given. Choose the alternative that is the d . d\' fhf . erivatlve, dx' 0 t e unction.

1. Y - (4x + 1)2(1 - x)'

(A) (4x + 1)2(1 - x)2(5 - 2Ox) (8) (4x + 1)(1 - x)2(4x + II) (C) 5(4x + 1)(1 - X)2(1 - 4x) (D) (4x + 1)(1 - x)2(l1 - 2Ox) (E) - 24(4x + 1)(1 - X)2

2-Jt 2. Y = h+l

7 6%-5 9 (A) - (3x + 1)2 (8) (3x + 1)2 (C) - (3x + 1)2

7 7-m-(0) (3.% + 1)2 (E) (3x + 1)2

3. Y - V3 - lx

1 (A) 2V3 - lx

(8) - 1 V3 - 2x

_ (3 - lx)J12 (C) 3

(0) - 3 ~ 2x (E) ~ (3 - 2x)J12

4. _ 2

Y - (5.r + 1)3

(A) 30 (8) - 30(5.% + 1)-4 (C) -6 - (5.% + 1)2 (5.% + It

(D) _ 10 (5x + 1)-4Il (E) 30 3 (5x + 1)4

5. y_~-4xJI2-2

2%113 - 2%-11l lx- l13 - 2%-11l (C) 9 (A) (8) - x'" - SxlIZ 5

(D) 2 2 (E) lx- l13 - 2x- 11l ----2 x l13 XII2

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