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JEE Main20258 Apr 2025Evening ShiftMathematicsLimitsActual

Given below are two statements : Statement I : _ x 0 ( ⁻¹ x+ _e 1+x 1-x -2 x x^5 )= 2 5 Statement II : _ x 1 ( x ^ 2 1- x )= 1 e ^2 In the light of the above statements, choose the correct answer from the options given below :

Options

  1. AStatement I is false but Statement II is true
  2. BStatement I is true but Statement II is false
  3. CBoth Statement I and Statement II are false
  4. DBoth Statement I and Statement II are true

Correct answer

D. Both Statement I and Statement II are true

Step-by-step solution

aligned & _ x 0 ⁻¹ x+ 1 2 [ (1+x)- (1-x)]-2 x x^5 & = _ x 0 (x- x^3 3 + x^5 5 )+ 1 2 [x- x^2 2 + x^3 3 - (-x- x^2 2 - x^3 3 ) ]-2 x x^5 & = _ x 0 2 x+ 2 x^5 5 -2 x x^5 = 2 5 & _ x 1 x^ 2 (1-x) =e^ _ x l ( 2 (1-x) )(x-1) =e⁻² aligned Both statements correct

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