Evaluate lim x→0 (cos x − 1)/x^2.

Evaluate lim x→0 (cos x − 1)/x^2.

• A. 0
• B. -1/2
• C. 1/2
• D. -1

Answer: (B)

When x is near zero, cos x can be approximated by its second-order Taylor expansion: cos x ≈ 1 - x^2/2. Subtracting 1 gives cos x − 1 ≈ −x^2/2. Dividing by x^2 yields (cos x − 1)/x^2 ≈ (−x^2/2)/x^2 = −1/2. As x approaches 0, the higher-order terms vanish, so the limit is −1/2. This matches the correct choice. The sign comes from the fact that cosine dips below 1 near 0, and the leading term is proportional to x^2 with a negative coefficient.

When x is near zero, cos x can be approximated by its second-order Taylor expansion: cos x ≈ 1 - x^2/2. Subtracting 1 gives cos x − 1 ≈ −x^2/2. Dividing by x^2 yields (cos x − 1)/x^2 ≈ (−x^2/2)/x^2 = −1/2. As x approaches 0, the higher-order terms vanish, so the limit is −1/2. This matches the correct choice. The sign comes from the fact that cosine dips below 1 near 0, and the leading term is proportional to x^2 with a negative coefficient.