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JEE Main202330 Jan 2023Evening ShiftMathematicsDefinite IntegrationActual

lim n → ∞ 3 n 4 + 2 + 1 n 2 + 2 + 2 n 2 + … + 3 - 1 n 2 is equal to

Options

  1. A12
  2. B19 3
  3. C0
  4. D19

Correct answer

D. 19

Step-by-step solution

Let L = lim n → ∞ 3 n 4 + 2 + 1 n 2 + 2 + 2 n 2 + . . . + 3 - 1 n 2 ⇒ L = lim n → ∞ 3 n 2 + 0 n 2 + 2 + 1 n 2 + 2 + 2 n 2 + . . . + 2 + 1 - 1 n 2 ⇒ L = 3 lim n → ∞ 1 n ∑ r = 0 n - 1 2 + r n 2 ⇒ L = 3 ∫ 0 1 2 + x 2 d x ⇒ L = 2 + x 3 0 1 ⇒ L = 27 - 8 ⇒ L = 19

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