Baixe 65878816 - Functions - Limits - amp - Continuity - Qns e outras Exercícios em PDF para Matemática, somente na Docsity!
QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
1. If f(x) = cos (log x), then
f(x) f(y) - 1
2 [^ (^ ) ]
f xy + f (x y ) (^) =
(A) - 1 (B) (^12) (C) - 2 (D) None of these
2. If f(x) =
x (^) x x x
sin , ,
, then
Limit x → 0 f(x) = (A) 1 (B) 0 (C) - 1 (D) None of these
3. The function,
f(x) = log ( 1 + a x) − log ( 1 −b x) x is not defined at x = 0. The value which should be assigned to f at x = 0, so that it is continuous at x = 0, is : (A) a - b (B) 1 + b (C) log a + log b (D) None of these
4. Let f(x) =
x x x x if x k if x
3 2 2
If f(x) be continuous for all x, then k is equal to : (A) 7 (B) - 7 (C) ± 7 (D) None of these
5. Limit x → 1 (1 - x) tan π^ x 2
^
^
(A) π 2 (B) π + 2
(C) 2
π (D) None of these
6. In order that the function, f(x) = (x + 1)1/x^ is continuous at x = 0, f(0) must be defined as : (A) f(0) = 0 (B) f(0) = e (C) f(0) = 1/e (D) f(0) = 1 7. Domain of the function,
sin l n
−^2
x x is :
(A) [- 2, 1] (B) (- 2, 1) (C) [- 2, 1) (D) (- 2, 1]
8. If f(9) = 9, f ′ (9) = 4, then
Limit x → 9 f x x
(A) 2 (B) 4
(C) - 2 (D) - 4
9. Limit h → 0 x h x h
(A) 1
2 x
(B)
x (C) (^2) x (D) (^) x
10. Limit x → 0
x x
( + ) /^ −
(A) log 2 (B) log 4 (C) log 2 (D) None of these
11. If f(x) =
x x x for x for x
2 2
then : (A) Limit x → 1 + 0 f(x) = 2 (B) Limit x → 1 − f(x) = 3 (C) f(x) is discontinuous at x = 1
Function, Limits & Continuity
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
(D) None of these
12. If f(x) =
sin x cos x x^ when x when x
then : (A) Limit x → 0 + f(x) ≠ 0 (B) Limit x → 0 −f(x) = 0 (C) f(x) is continuous at x = 0 (D) None of these
13. Limit x → π 4 sin α cosα α π
(A) 2 (B)
(C) 1 (D) None of these
14. Limit x → π 2 tan x log sin x =
(A) 0 (B) 1 (C) - 1 (D) None of these
15. Limit x → 0 tan sin
x x x x
(A) 0 (B) 1
(C) 1
(D) 1
16. Limit x → 0 cos^ a x^ cosb x x
(A) a^ b
2 2 2
− (^) (B) b 2 a^2 2
(C) a^2 - b^2 (D) b^2 - a^2
17. If f(x) =
x
x
x x x
, then :
(A) Limit x → 0 + f(x) = 1
(B) Limit x → 0 − f(x) = 1 (C) f(x) is discontinuous at x = 0 (D) None of these
18. The value of Limit x → ∞^ x^ bx x ax
2 2
is
(A)
b a
(B) 1
(C) 0 (D) 4
19. Limit x → 0
x
a x bx
(A) log b a
^
^
(B) log a b
^
(C) a b (D) log ab
20. If f(x) =
[ ]
[ ]
sin (^) , [ ] , [ ]
x x when^ x 0 when^ x
where
[x] is greatest integer function, then Limit x → 0 f(x) = (A) - 1 (B) 1 (C) 0 (D) None of these
21. Limit x → 0 sin x x x
(A) 1
(B) - 1
(C) 1
(D) - 1
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
(B) f(x) is continuous at x = π 2 (C) f(x) is continuous at x = 0 (D) None of these
32. If f(x) =
( )
1 4
16 4
−
cos (^) , , ,
x x x x
a
when x when x when x
is continuous at x = 0, then the value of ‘a’ will be : (A) 8 (B) - 8 (C) 4 (D) None of these
33. Domain of the function,
f(x) = x x x
is :
(A) (1, 2) (B) (− ∞, − 2) ∪ (2, ∞) (C) (− ∞, − 2) ∪ (1, ∞) (D) (− ∞, ∞) − (1, ± 2)
34. Limit h → 0
[ (^ )^ (^ )]
sin cos cos sin
π (^) + − π+
−
h h h h h
(A) - 2
(B) - 3
(C) - 2 3 (D) 4
35. If f(x) =
a x b
x
when x when x when x
2 2 1
is
continuous at x = 1, then the most suitable value of a, b are : (A) a = 2, b = 0
(B) a = 1, b = - 1 (C) a = 4, b = 2 (D) All the above
36. If function f(x) = 1 2 - tan π x 2
^
^ ;
(− 1 < x < 1) & g (x) = (^3) + 4 x − 4 x^2 , then the domain of gof is :
(A) (- 1, 1) (B) −
^
(C) −
^
, (D) − −
^
37. Limit x → ∞^ x e
n x =^0 such^ that^ n^ is^ an integer for : (A) No value of n (B) All values of n (C) Only negative values of n (D) Only positive values of n
38. If f(x) =
x x x when x when x
^2
, then
(A) f(x) is continuous at x = 0 (B) f(x) is discontinuous at x = 0 (C) Limit x → 0 f(x) = 2 (D) None of these
39. If f(x) = x x
when x when x
2 5
+^1
then
(A) f(x) is continuous at x = 1 (B) f(x) is discontinuous at x = 1 (C) Limit x → 1 f(x) = 1 (D) None of these
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
40. If f(x) =
x x x x a
when x when x
2 2
is continuous at x = - 5, then the value of ‘a’ will be :
(A) 3 2
(B) 7
(C)
(D)
41. If f : R → R be a diff. function and f(1) = 4, then the value of,
Limit x → 1 2 4 1
t x
f x ∫ −
( ) dt =
(A) 8 f ′ (1) (B) 4 f ′ (1) (C) 2 f ′ (1) (D) f ′ (1)
42. If f(a) = 2, f ′ (a) = 1, g(a) = 1, g ′ (a) = 2,
then Limit x →a^ g x^ f a^ g a^ f x x a
(A) 1
(B) - 1
(C) 3 (D) - 3
43. The range of f(x) = cos x - sin x, is (A) (- 1, 1) (B) [- 1, 1)
(C) − ^
π π 2 2
, (D) [ −^2 ,^2 ]
44. If f(x) =
x
x
x x x
λ 4 3 4
is continuous
at x = 3, then λ = (A) 4 (B) 3 (C) 2 (D) 1
45. Let,
f(x) =
( 1 ) 6 0 0 (^0 ) 2 3
sin , , ,
sin
tan tan
x b e
x x x
a x
x x
π
π
then the values of a & b if f is continuous at x = 0, are respectively
(A) 2 3
, (B)
, e2/
(C) 3
, e3/2^ (D) None of these
46. Limit x → ∞
x x
+^ x
3 is :
(A) 1 (B) e (C) e^2 (D) e^3
47. Let function f(x) = x^2 + x + sin x - cos x + log (1 + x) be defined over the interval [0, 1]. The odd extentions of f(x) to interval [- 1, 1] is : (A) x^2 + x + sin x + cos x − log (1 + x) (B) − x^2 + x + sin x + cos x − log (1 + x) (C) − x^2 + x + sin x − cos x − log (1 + x) (D) None of these 48. The value of,
Limit n → ∞^ n n
n n
n 1 4 9 n n
is equal to :
(A) π 2 (B) π 4 (C) 1 (D) None of these
49. Limit n → ∞
n n n
..... (^) is
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
(A) 99
(B) 1
(C) 1
(D) 1
62. If f(x) = sin , , ,
x x n n Z 2 otherwise
^ ≠^ ∈
π and
g(x) =
x x x x
, then
Limit x → 0 g {f(x)} is : (A) 5 (B) 6 (C) 7 (D) 1
63. The values of a & b such that,
Limit x → 0 x^ a^ x^ b^ x x
( 1 cos ) sin 3
(A)
(B) 5 ,
(C) − 5 −
, (D) None of these
64. If f(x) = (a - xn)1/n, where a > 0 and n is a positive integer, then f [f(x)] = (A) x^3 (B) x^2 (C) x (D) None of these 65. If f is an even function defined on the interval (- 5, 5), then four real values of x satisfying the equation,
f(x) = f x x
^
are :
(A) −^3 −^5 −^ +^ −^ +
(B) −^5 +^3 −^ +^ +^ −
(C) 3 5
(C) − 3 − 5 , − 3 + 5 , 3 − 5 , 3 + 5
66. Let f(x) = [x] sin π [ x + ]
^
, where [.]
denotes the greatest integer function. The domain of f is ______ and the points of discontinuity of f in the domain are : (A) {x ∈ R x ∈ [− 1, 0)} , I − {0} (B) {x ∈ R x ∉ [1, 0)} , I − {0} (C) {x ∈ R x
[− 1, 0)} , I − {0}
(D) None of these
67. The inverse of the function,
e ef(x) = e e
x x x x
− −
(A) loge^ x x
1 2/ (B) loge^ x x
1 2/
(C) loge^ x 2 x
1 2 −
/ (D) loge^ x x
2
68. If the domain of function, f(x) = x^2 - 6x + 7 is (− ∞, ∞), then the range of function is : (A) (− ∞, ∞) (B) (− 2, ∞) (C) (− 2, 3) (D) (− ∞, − 2) 69. If f(x) = x x − 1 , then f a f a
(A) f (- a) (B) f (1/a)
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QUEST TUTORIALS Head Office : E-16/289, Sector-8, Rohini, New Delhi, Ph. 65395439
(C) f(a^2 ) (D) f
^
a a 1
70. Function f(x) = x x x x
is a
continuous function : (A) For all real values of x (B) For x = 2 only (C) For all real values of x such that x ≠ 2 (D) For all integral values of x only
ANSWERS
1. D 2. B 3. B 4. A 5. C 6. B
- B 8. A 9. A 10. B 11. C 12.C
- A 14. A 15. C 16. B 17. C 18.B
- B 20. D 21. D 22. C 23. B 24.A
- C 26. C 27. B 28. B 29. B 30.C
- A 32. A 33. B 34. D 35. D 36.A
- B 38. B 39. B 40. B 41. A 42.C
- D 44. D 45. B 46. B 47. B 48.B
- B 50. C 51. D 52. C 53. A 54.A
- B 56. B 57. C 58. A 59. A 60.D
- B 62. D 63. C 64. C 65. A 66.C
- B 68. B 69. C 70. A
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