Evaluate the limit as x approaches 0 of (1 - cos x) / x.

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Multiple Choice

Evaluate the limit as x approaches 0 of (1 - cos x) / x.

Explanation:
Near zero, cos x can be approximated by its Taylor expansion: cos x ≈ 1 - x^2/2, so 1 - cos x ≈ x^2/2. Dividing by x gives (1 - cos x)/x ≈ x/2, which tends to 0 as x → 0. Therefore the limit is 0. Another way to see it is by L’Hôpital’s rule: differentiating top and bottom gives sin x, which tends to 0 as x → 0.

Near zero, cos x can be approximated by its Taylor expansion: cos x ≈ 1 - x^2/2, so 1 - cos x ≈ x^2/2. Dividing by x gives (1 - cos x)/x ≈ x/2, which tends to 0 as x → 0. Therefore the limit is 0. Another way to see it is by L’Hôpital’s rule: differentiating top and bottom gives sin x, which tends to 0 as x → 0.

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