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JEE MainMathematicsLimits

If _ x 1/2 a ( ⁻¹x) - x 1 - 3 ( ⁻¹x) = L , where L is a finite real number, then the ordered pair (a, L) is equal to

Options

  1. A( 3 , 1 2 )
  2. B( 1 3 , - 1 2 )
  3. C( 3 , - 1 2 )
  4. D( 1 3 , 1 2 )

Correct answer

D. ( 1 3 , 1 2 )

Step-by-step solution

Let t = ⁻¹x , which implies x = t . As x 1/2 , t /3 . The limit becomes: _ t /3 a t - t 1 - 3 t = L As t /3 , the denominator approaches 1 - 3 ( /3) = 1 - 3 ( 1 3 ) = 0 . For the limit L to be finite, the numerator must also approach 0 as t /3 . a ( /3) - ( /3) = 0 a ( 3 2 ) - 1 2 = 0 a = 1 3 Substitute a = 1 3 back into the limit: L = _ t /3 1 3 t - t 1 - 3 t t L = _ t /3 t - 3 t 3 t - 3 t t L = _ t /3 t 3 L = ( /3) 3 = 3 /2 3 = 1 2 Thus, (a, L) = ( 1 3 , 1 2 ) . Answer: ( 1 3 , 1 2 )

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