Evaluate lim_{x→0} (1 - cos x)/x^2.

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Multiple Choice

Evaluate lim_{x→0} (1 - cos x)/x^2.

Explanation:
Recognize how small angles behave and use a trig identity to simplify. The identity 1 − cos x = 2 sin^2(x/2) lets you rewrite the expression as (2 sin^2(x/2))/x^2, which is (1/2) [sin(x/2)/(x/2)]^2. As x approaches 0, x/2 also approaches 0, and the standard limit sin y / y → 1 gives sin(x/2)/(x/2) → 1. Therefore the limit is (1/2) · 1^2 = 1/2.

Recognize how small angles behave and use a trig identity to simplify. The identity 1 − cos x = 2 sin^2(x/2) lets you rewrite the expression as (2 sin^2(x/2))/x^2, which is (1/2) [sin(x/2)/(x/2)]^2. As x approaches 0, x/2 also approaches 0, and the standard limit sin y / y → 1 gives sin(x/2)/(x/2) → 1. Therefore the limit is (1/2) · 1^2 = 1/2.

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