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AP EAMCET202120 Aug 2021Evening ShiftMathematicsLimitsActual

lim x → 1 ( 1 - x ) 1 - x 2 ⋯ ⋯ 1 - x 2 n ( 1 - x ) 1 - x 2 ⋯ ⋯ 1 - x n 2 = ¯ , ∀ n ∈ N

Options

  1. AP n 2 n
  2. BC n 2 n
  3. C( 2   n )   !
  4. D( 2 n ) ! n !

Correct answer

B. C n 2 n

Step-by-step solution

lim x → 1 ( 1 - x ) 1 - x 2 ⋯ ⋯ 1 - x 2 n ( 1 - x ) 1 - x 2 ⋯ ⋯ 1 - x n 2     ∀ n ∈ N lim x → 1 ( 1 - x ) 1 - x 2 ⋯ ⋯ 1 - x 2 n 1 - x 2 n ( 1 - x ) 1 - x 2 ⋯ ⋯ 1 - x n 2 1 - x 2 n lim x → 1 1 - x 1 - x × 1 - x 2 1 - x . . . . . . . . . . . . . × 1 - x 2 n 1 - x 1 - x 1 - x × 1 - x 2 1 - x × . . . . . . . . . . . . . . . . . × 1 - x n 1 - x 2 Apply L'hospital rule lim x → 1 - 1 - 1 × - 2 x

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