What is the limit as x -> 0 of (1 - cos x)/x^2?

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Multiple Choice

What is the limit as x -> 0 of (1 - cos x)/x^2?

Explanation:
This limit hinges on a standard small-angle connection between sine and its ratio to its argument. Use the identity 1 − cos x = 2 sin^2(x/2). Then (1 − cos x) / x^2 = [2 sin^2(x/2)] / x^2. Let y = x/2, so this becomes (1/2) [sin(y)/y]^2. As x → 0, y → 0, and sin(y)/y → 1. Therefore the limit is (1/2) · 1^2 = 1/2.

This limit hinges on a standard small-angle connection between sine and its ratio to its argument. Use the identity 1 − cos x = 2 sin^2(x/2). Then

(1 − cos x) / x^2 = [2 sin^2(x/2)] / x^2. Let y = x/2, so this becomes (1/2) [sin(y)/y]^2.

As x → 0, y → 0, and sin(y)/y → 1. Therefore the limit is (1/2) · 1^2 = 1/2.

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