Objective - Lcd

FIITJEE CHENNAI CENTRE. 175 POONNAMMALLEE HIGH ROAD. OPP. EGA THEATRE, KILPAUK. CHENNAI-10 PH. 044 - 42859701 (MNT)

LIMITS, CONTINUITY

AND DIFFERENTIABILITY

NAME: ________________________

BATCH CODE – MDIT13A01

PRACTICE SHEET - LCD FIITJEE CHENNAI CENTRE


1. The number of critical points of f (x) = max (sin x , cos x) for x (0 , 2 )

(A) 2 (B) 5 (C) 3 (D) non

2. If f (x) =

x

0

)1t( (et –1) (t – 2) (t + 4) dt then f (x) would assume the local

minima at;

(A) x = - 4 (B) x = 0

(C) x = -1 (D) x = 2.

3.

x2

xcos6Lim

2x

, where [.] denotes the greatest integer function, is equal to;

(A) - 3 (B) - 4 (C) -2 (D) none of these

4. Let f(x) = 4

x

1

xtan

x (0, /2) ~ {/4}, then the value of f(/4} such that f(x) becomes

continuous at x = 4

is equal to;

(A) e (B) e (C) e

1 (D) e2

5. Let f(x)=[5+3 sinx] x R. Then total number of points of discontinuity of f(x) in [0, ] is equal to;

(A) 5 (B) 6 (C) 7 (D) 4

6. f(x) = sin-1(sinx), x [-2, 2]. Total number of critical points of f(x) is ; (A) 3 (B) 4 (C) 5 (D) 2 7. If the line ax + by + c = 0 is normal to the curve x y + 5 = 0 then

(A) a > 0 , b > 0 (B) b > 0 , a < 0

(C) a < 0 , b < 0 (D) b < 0 , a > 0

8. The maximum value of f(x) = |x ln x| in x(0,1) is;

(A) 1/e (B) e (C) 1 (D) none of these

9. f(x) = 3x3 +4ex – k is always increasing then value of k = (A) 2 (B) –4/9 (C) 4/9 (D) all of these

10. x]2x[]x2[lim2x

is

(A) 0 (B) 3 (C) –3 (D) does not exist

11. Let f (x) be a twice differentiable function and f (0) = 2 then

22x x

x4fx2f3xf2lim

is

(A) 6 (B) 1 (C) 12 (D) 3



12 Let h (x) = f (x) –{f (x)}2 + {f (x)}3 for all real values of x then

(A) h is whenever f (x) is (B) h is whenever f(x) is 0

(C) h is whenever f is (D) nothing can be said in general

13. Let f (x) > 0, g (x) < 0 for all x R, then (A) f {g (x)} > f {g (x + 1)} (B) f {g (x)} > f {g (x –1)} (C) g {f (x)} >< g {f (x + 1)} (D) g {f (x)} > g {f (x –1)}

14. xl

xlnlimx

= ………………………………………………………….. [.] G. I. F

15. n1

nnn

n753lim

= ……………………………………………

16. If , are the roots of ax2 + bx + c = 0 then 2

2

x x

cbxaxcos1lim

= ………………………

17. x11xx1lim1x

= ………………………………………………………………

18. f (x) = sin-1(cos x) then points of nondifferentiability between [0, 2] = ……………………..

19. Let f (x + y) = f (x) . f (y) for all x & y, if f (5) = 2 and f’(0) = 3, then f’ (5) = …………………….

20. f(x) =

2x,2x

]x[x

2x,b

2x,xx2

|2xx|a2

2

( where [.] denotes the greatest integer function ). If f(x)

is continuous at x = 2, then (A) a = 1, b = 2 (B) a = 1, b = 1 (C) a = 0, b = 1 (D) a = 2, b = 1

21. Let f(x) =

0x,1

0x,0

0x,1

and g(x) sinx + cosx, then points of discontinuity of f{g(x)} in (0,

2) is

(A)

4

3,

2 (B)

4

7,

4

3

(C)

3

5,

3

2 (D)

3

7,

4

5

22. If and are the roots at ax2 + bx + c = 0 then

x/12

xcbxax1lim is

(A) a ( – ) (B) ln|a( – )|

(C) ea( – ) (D) ea| – |



23. 4

x

lim

1xcot

1xcos2

is equal to

(A) 2/1 (B) 1/2

(C) 22

1 (D) 1

24. The function f(x) = [x]2 – [x2] where [y] denotes the greatest integer less than or equal to y),

is discontinuous at (A) all integers (B) all integers except 0 and 1 (C) all integers except 0 (D) all integers except 1

25. If the derivative of f(x) w.r. t x is xf

xsin2

1 2

, then f(x) is a periodic function with period

(A) (B) 2

(C) /2 (D) none of these.

26.

x

xsin))7y2y(min(lim 2

0x = ? (where [.] denotes greatest integer function)

(A) 4 (B) 5 (C) 6 (D) none of these

27.

x

xtan100lim

0x = ? (where [.] denotes greatest integer function)

(A) 100 (B) 99 (C) 101 (D) 0

28. If f (x) = |cos 2x|, then f

0

4 is equal to

(A) 2 (B) 0 (C) –2 (D) doesn’t exist

29. xcos/1

2/x)x(sinlim

=

(A) 0 (B) e (C) 1 (D) doesn’t exist

30. 40x x

)xcos1cos(1lim

equals to

(A) 2

1 (B)

8

1 (C)

4

1 (D)

16

1

31. xtanIn

1

xtan2lim4/x

equals to

(A) e (B) 1 (C)0 (D) e–1

32.

2

2

0x x

xcossinlim

equals to

(A) 0 (B) (C) – (D) not exist



33. If tan-1 (x +h) = tan-1(x) + (h siny)(siny) – (h siny)2 . 2

y2sin + (h siny)3.

3

y3sin + . .. .,

where x ( 0, 1), y (/4, /2) , then (A) y = tan-1x (B) y = sin-1x (C) y = cot-1x (D) y = cos-1x

34. The value of ]))x(sin(tantancos[lim 11

x

is equal to

(A) -1 (B) 2 (C) 2

1 (D)

2

1

35. If

x

0

2

0x1

taxsinx

dttlim , then the value of a is

(A) 4 (B) 2 (C) 1 (D) none of these 36. For some g, let f(x) = x(x+3) eg(x) be a continuous function. If there exists only one point x = d

such that f(d) = 0, then

(A) d < -3 (B) d > 0 (C) -3 d 0 (D) -3 <d < 0

37.

1n

n n

11ln1lim is equal to

(A) 0 (B) 1 (C) e (D) none of these

38. The value of ]x[

1nn

x e

1nxxlim

, n I is

(A) 1 (B) 0 (C) n (D) n(n –1)

39. Given a function f(x) continuous x R such that

xflog

e

11logxflim

xf0x = 0,

then f(0) is (A) 0 (B) 1 (C) 2 (D) 3

40. Let R be the set of real numbers and f : R R be such that for all x and y in R

| f (x) – f (y) | | x –y |7. Then f (x) is. (A) linear (B) constant (C)quadratic (D) none of these.

41. Find the value of

xcot

x

1lim 2

20x

(A) 2/5 (B) 2/3 (C) 1/4 (D) 1/5.

42.

x

x2cos12

1

lim0x

is

(A) 1 (B) –1 (C) 0 (D) doesn’t exist



43. Given that f (x) is a non-zero differentiable function such that f (x + y) = f (x). f (y), x, y R,

and f (0) = 1 then ln f (1) is equal to

(A) 0 (B) 1

(C) e (D) none of these

44. The largest interval where the function f (x) = |x|1

x

is differentiable

(A) (–, ) (B) (0, )

(C) (–, 0) (0, ) (D) none of these

45. The value of ylimy

ln

xsin

)y/1xsin(, when 0 < x < / 2, is

(A) cos x (B) (C) cot x (D) does not exist 46. f(x) = [x] + |x–1| then f(x), (where [.] denotes greatest integer function) is (A) Continuous x = 0 (B) not differentiable at x = 0. 5 (C) Discontinuous at x = 2 (D) differentiable at x = – 2

47. f(x) = [cos x + sin x] , 0 < x < 2 the number of points of discontinuity of f(x) is (A) 6 (B) 5 (C) 3 (D) 4

48. g(x) = x |x| then g (x) is (A) Does not exist at x = 0 (B) always positive (C) Always none negative (D) always none zero

49. f(x) = 1]x[

])x[sin(2

, (where [.] denotes greatest integer function) then f (x) is

(A) Discontinuous at x = (B) differentiable function for all x (C) f’ (X) does not exist (D) none of these

50. If f(x) =

x

2

2

2xx

3x5x

, then xflim

x is

(A) e4 (B) e3 (C) e2 (D) 24

51. If f(x + y) = f(x). f (y) for all x and y and f(5) = 2, f (0) = 3, then f(5) will be (A) 2 (B) 4 (C) 6 (D) 8 52. If f (x) is differentiable every where than

(A) )x(f is differentiable every where

(B) )x(f2 is differentiable every where

(C) f (x) / )x(f is differentiable every where

(D) none of these



53. The value of

2

23

x x31

x2x/1sinxlim

(A) is 0 (B) is -1/3

(C) is –1 (D) is -2/3

54. The set of all point, where the function f(x) = x1

x

is differentiable, is

(A) (–, ) (B) (0, )

(C) (–, 0)(0, ) (D) none of these

55. If f (x) =

0xfor0

0xforx

xsin

where [x] denotes greatest integer function, then xflim0x

=

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

56. Let f (x) be a differentiable function and f (1) = 2. If

)x(f

21x

dt1x

t2lim = 4, then value of f (1) is

(A) 1 (B) 2 (C) 4 (D) none of these

57. 2

2

0x x

)xcossin(lim

is equal to

(A) – (B)

(C) 2

(D) 1

58. If {x} denotes the fractional part of x, then. x

1elim

}x{

0z

is

(A) 0 (B) 1

(C) (D) none of these 59. Let h (x) = max. {–x, 1, x2 }, for every real x. then number of points of non – differentiability is (A) 1 (B) 2 (C) 3 (D) 4

60. Let f (x) =

0x,0

0x,

x

1sinx

1sin

x

1sinx

, then f(x) is

(A) both continuous & differentiable at x = 0 (B) continuous but not differentiable at x = 0 (C) neither continuous nor differentiable at x = 0 (D) f’ (0 – 0) exists.

Objective - Lcd

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