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The value of l i m x → ∞ 1 1 x + 2 1 x + 3 1 x + . . . + 1 0 1 x 10 10 x is

Options

  1. A10 !
  2. B10
  3. C9 !
  4. D0

Correct answer

A. 10 !

Step-by-step solution

l i m x → ∞ 1 1 x + 2 1 x + 3 1 x + . . . + 1 0 1 x 10 10 x = l i m y → 0 + 1 y + 2 y + 3 y + . . . + 1 0 y 10 10 y w h e r e, y = 1 x = e l i m y → 0 + 10 y 1 y + 2 y + 3 y + . . . + 1 0 y 10 - 1 = e l i m y → 0 + 1 y + 2 y + 3 y + . . . + 1 0 y - 10 y = e l i m y → 0 + 1 y - 1 y + 2 y - 1 y + 3 y - 1 y + . . . + 1 0 y - 1 y = e l o g 1 + l o g 2 + l o g 3 + . . . + l o g 10 = e l o g 1 ⋅ 2 ⋅ 3 . . . . . 10 = 10 !

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