Let fx + y = fxfy and fx&nbs

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11.

Let fx+ y = fxfy and fx = 1 + xgxϕx, wherelimx0gx = a and limx0ϕx = b. Then what is f'x = ?

  • 1 + abf(x)

  • 1 +ab

  • ab

  • abf(x)


D.

abf(x)

f'x = limh0fx + h - fxh       = limh0fxfh - fxh       = limh0fxfh - 1h       = limh0fxhghϕhh       = abfx


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12.

What is limx0ex - 1 - xx2 = ?

  • 0

  • 12

  • 1

  • 2


13.

What is 0π2dθ1 + cosθ = ?

  • 12

  • 1

  • 3

  • None of the above


14.

If fx = 9 - x2, then what is limx1fx - f1x - 1 = ?

  •  - 142

  • 18

  • - 122

  • 122


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15.

Consider the following statements : 

  1. If limxafx and limxagx both exist, then limxafxgx exists
  2. If  If limxafxgx, then both limxafx and limxagx must exist.

Which of the above statements is/are correct?

  • 1 only

  • 2 only

  • both 1 and 2

  • neither 1 nor 2


16.

Consider the following statements:

  1. Derivative of f(x) may not exist at some point.
  2. Derivative of f(a) may exist finitely at some point
  3. Derivative of f(x) may be infinite (geometrically) at some point.

Which of the above statements are correct ?

  • 1 and 2 only

  • 2 and 3 only

  • 1 and 3 only

  • 1, 2 and 3


17.

If limxπ2sinxx = l and limxcosxx = m, then which one of the following is correct ?

  • l = 1, m = 1

  • l = 2π, m = 

  • l = 2π, m = 0

  • l = 1, m = 


18.

The left-hand derivative of f(X) = [x] sinπx at x = k

where k is an integer and [x] is the greatest integer function, is

  •  - 1kk - 1π

  •  - 1k - 1k - 1π

  •  - 1k

  •  - 1k - 1kπ


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19.

Consider the function

fx = ax + x - 1x + xwhere  .  denotes the greatest integer functionWhat is limx0fx = ?

  • 1

  • ln(a)

  • 1 - a - 1

  • Limit does not exist


20.

Consider the function

fx = ax + x - 1x + xwhere  .  denotes the greatest integer functionWhat is limx0-fx = ?

  • 0

  • ln(a)

  • 1 - a - 1

  • Limit does not exist


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