Evaluate the limit as x approaches 0 of (1 - cos 3x) / x^2.

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Multiple Choice

Evaluate the limit as x approaches 0 of (1 - cos 3x) / x^2.

Explanation:
Small-angle behavior of cosine: when t is very small, cos t ≈ 1 − t^2/2, so 1 − cos t ≈ t^2/2. Here we take t = 3x, giving 1 − cos 3x ≈ (3x)^2/2 = 9x^2/2. Dividing by x^2 leaves 9/2, so the limit as x approaches 0 is 9/2.

Small-angle behavior of cosine: when t is very small, cos t ≈ 1 − t^2/2, so 1 − cos t ≈ t^2/2. Here we take t = 3x, giving 1 − cos 3x ≈ (3x)^2/2 = 9x^2/2. Dividing by x^2 leaves 9/2, so the limit as x approaches 0 is 9/2.

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