The value of lim x → 0 + cos − 1 ⁡ ( x ) ⋅ sin − 1 ⁡ ( x ) x ( 1 − x 2 ) , where [x] denotes the greatest integer ≤ x isA) 0B) π 4 C) π 2 D) π

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The value of       lim          x      →              0        +                                cos                  −          1                    ⁡      (      x      )      ⋅              sin                  −          1                    ⁡      (      x      )              x      (      1      −              x        2            )      , where [x] denotes the greatest integer   ≤  x isA) 0B)       π    4  C)       π    2  D)   π

Sailiaadvadesw95 · Accepted Answer

The correct option is C       π    2  Find the value of the given expression:Given:       lim          x      →              0        +                                cos                  −          1                    ⁡      (      x      )      ⋅              sin                  −          1                    ⁡      (      x      )              x      (      1      −              x        2            )      The greatest integer function is denoted by [x] for any real function.The function rounds – off the real number down to the integer less than the number.Thus, the value of [x] in given expression is 0.Because for any number between one and zero value of [x] is zero.limx→0+cos-1(x).sin-1(x)x(1–x2)=limx→0+cos-1(x)(1–x2)×.sin-1(x)x=limx→0+cos-1(x)(1–x2)×1∵limx→0sin-1xx=1=limx→0+cos-1(0+)(1–0+2)=π2∵cos-10=π2Therefore, answer (C) i.e.       π    2   is correct.

The value of lim_{x -> 0^+} (cos^{-1}(x)cdot sin^{-1}(x))/(x(1-x^2)), where [x] denotes the greatest integer <= x

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