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

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Multiple Choice

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

Explanation:
For small x, cos(2x) behaves like 1 minus (2x) squared over 2, so 1 - cos(2x) is about 2x^2. Dividing by x^2 gives a limit of 2. A clean way to see this exactly is to use the identity 1 - cos(2x) = 2 sin^2(x). Then the expression becomes 2 (sin x / x)^2, and since sin x / x → 1 as x → 0, the limit is 2.

For small x, cos(2x) behaves like 1 minus (2x) squared over 2, so 1 - cos(2x) is about 2x^2. Dividing by x^2 gives a limit of 2. A clean way to see this exactly is to use the identity 1 - cos(2x) = 2 sin^2(x). Then the expression becomes 2 (sin x / x)^2, and since sin x / x → 1 as x → 0, the limit is 2.

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