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

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

• A. 0
• B. 1/2
• C. 1
• D. 3/4

Answer: (D)

When x is very small, cosine can be expanded as cos x = 1 - x^2/2 + higher-order terms. So 1 - cos x ≈ x^2/2. Dividing by x^2 gives a value approaching 1/2 as x → 0. Therefore, the limit is 1/2. If you prefer a calculus route, you can use L'Hôpital's rule: differentiate top and bottom to get sin x / (2x), which tends to (1/2) as x → 0 because sin x / x → 1.

When x is very small, cosine can be expanded as cos x = 1 - x^2/2 + higher-order terms. So 1 - cos x ≈ x^2/2. Dividing by x^2 gives a value approaching 1/2 as x → 0. Therefore, the limit is 1/2.

If you prefer a calculus route, you can use L'Hôpital's rule: differentiate top and bottom to get sin x / (2x), which tends to (1/2) as x → 0 because sin x / x → 1.