The value of lim x → ∞ ⁡ 2 x 1 / 2 + 3 x 1 / 3 + 4 x 1 / 4 + . . . . . + x 1 / n 2 x - 3 1 / 2 + 2 x - 3 1 / 3 + . . . . + 2 x - 3 1 / n is

Step-by-step solution

Given, lim x → ∞ ⁡ 2 x 1 / 2 + 3 x 1 / 3 + . . . . + n x 1 / n 2 x - 3 1 / 2 + 2 x - 3 1 / 3 + . . . + 2 x - 3 1 / n = lim h → 0 ⁡ 2 1 h 1 / 2 + 3 1 h 1 / 3 + . . . . + n 1 h 1 / n 1 h 1 / 2 2 - 3 h 1 / 2 + 1 h 1 / 3 2 - 3 h 1 / 3 + . . . . . + 1 h 1 / n 2 - 3 h 1 / n O n p u t t i n g x = 1 h a s x → ∞ , h → 0 = lim h → 0 ⁡ 2 + 3 h 1 2 - 1 3 + 4 h 1 2 - 1 4 + . . . . + n h 1 2 - 1 n 2 - 3 h 1 / 2 + h 1 2 - 1 3 2 - 3 h 1 / 3 + . . . . + h 1 2 - 1 n 2 - 3 h 1 / n = 2 + 0 + 0 + 0 + . . . . . . . 2 1 / 2 + 0 + 0 + . .