Evaluate lim x→0 (cos x − 1)/x^2.

Evaluate lim x→0 (cos x − 1)/x^2.

• A. -1/2
• B. 0
• C. 1/2
• D. -1

Answer: (A)

Near zero, cosine behaves like its quadratic approximation: cos x = 1 − x^2/2 + higher-order terms. Substituting into the expression gives (cos x − 1)/x^2 ≈ (−x^2/2)/x^2 = −1/2, and the higher-order terms vanish as x → 0. So the limit is −1/2. Another quick route is L’Hôpital’s rule: differentiate top and bottom to get (−sin x)/(2x), and as x → 0 this tends to −(1/2)·(sin x/x) = −1/2 since sin x/x → 1.

Near zero, cosine behaves like its quadratic approximation: cos x = 1 − x^2/2 + higher-order terms. Substituting into the expression gives (cos x − 1)/x^2 ≈ (−x^2/2)/x^2 = −1/2, and the higher-order terms vanish as x → 0. So the limit is −1/2.

Another quick route is L’Hôpital’s rule: differentiate top and bottom to get (−sin x)/(2x), and as x → 0 this tends to −(1/2)·(sin x/x) = −1/2 since sin x/x → 1.