Evaluate the limit: lim_(x -> 0) [(1 - cos(x)) / x^2].
Answer and explanation
Correct answer: D. 1/2
- Step 1: Direct substitution yields 0/0.
- Step 2: Apply L'Hôpital's Rule once: lim_(x -> 0) [sin(x) / (2x)].
- Step 3: Recall standard limit lim_(x -> 0) [sin(x) / x] = 1.
- Step 4: Evaluate: (1/2) * 1 = 1/2.
Answer
1/2.
Why not the others: '0' assumes numerator goes to 0 faster than x^2; '1' overlooks the denominator factor 2; '1/4' uses an incorrect trigonometric half-angle identity.
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