BITSAT2019MathematicsDefinite IntegrationActual
Suppose the limit L = lim n → ∞ n ∫ 0 1 1 1 + x 2 n d x exists and is larger than 1 2 , then
Options
- A1 2 < L < 2
- B2 < L < 3
- C3 < L < 4
- DL ≥ 4
Correct answer
A. 1 2 < L < 2
Step-by-step solution
We have, L = lim n → ∞ n ∫ 0 1 d x 1 + x 2 n We know that, 1 + x 2 n > 1 + n x 2 ⇒   1 1 + x 2 n < 1 1 + n x 2 ⇒   ∫ 0 1 d x 1 + x 2 n < ∫ 0 1 1 1 + n x 2 d x ⇒   ∫ 0 1 1 1 + x 2 n < 1 n tan - 1 n x 0 1 = 1 n tan - 1 n L = lim n → ∞ n ∫ 0 1 1 1 + x 2 n d x < lim n → ∞ n 1 n tan - 1 n ∴   L = π 2