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l i m x → π 2 1 - t a n x 2 1 - s i n x 1 + t a n x 2 π - 2 x 3 is equal to

Options

  1. A1 8
  2. B0
  3. C1 32
  4. D∞

Correct answer

C. 1 32

Step-by-step solution

l i m x → π 2 t a n π 4 - x 2 1 - s i n x π - 2 x 3 Let, x = π 2 + y ⇒ l i m y → 0 = t a n - y 2 1 - c o s y - 2 y 3 = l i m y → 0 - t a n y 2 ⋅ 2 s i n 2 y 2 - 8 y 3 = l i m y → 0 1 32 t a n y 2 y 2 ⋅ s i n y 2 y 2 2 = lim y → 0 1 32 × tan y 2 ( y 2 ) . lim y → 0 sin y 2 y 2 2 ∵ lim x → 0 sin x x = lim x → 0 tan x x = 1 ⇒ 1 32 × 1 × 1 2 = 1 32

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