Evaluate the limit: L = lim_(x -> 0) [x - sin(x)] / [x^3].

Answer and explanation

Correct answer: D. 1/6

  1. Step 1: Form is 0/0. Using Taylor expansion: sin(x) = x - x^3 / 3! + O(x^5) = x - x^3 / 6 + O(x^5).
  2. Step 2: Substitute into numerator: x - sin(x) = x^3 / 6 - O(x^5).
  3. Step 3: Divide by x^3: lim_(x -> 0) [x^3/6] / x^3 = 1/6.

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