Evaluate lim_{x->0} (1 - cos x)/x^2.

Evaluate lim_{x->0} (1 - cos x)/x^2.

• A. 0
• B. 1/2
• C. 1
• D. -1/2

Answer: (B)

Cosine behaves like 1 minus a quadratic term for small angles, so cos x ≈ 1 − x^2/2 as x → 0. That means 1 − cos x ≈ x^2/2, and dividing by x^2 leaves 1/2 in the limit. A clean way to see this uses the identity 1 − cos x = 2 sin^2(x/2): (1 − cos x)/x^2 = (1/2)[sin(x/2)/(x/2)]^2, and since sin t/t → 1 as t → 0, the limit is 1/2.

Cosine behaves like 1 minus a quadratic term for small angles, so cos x ≈ 1 − x^2/2 as x → 0. That means 1 − cos x ≈ x^2/2, and dividing by x^2 leaves 1/2 in the limit. A clean way to see this uses the identity 1 − cos x = 2 sin^2(x/2): (1 − cos x)/x^2 = (1/2)[sin(x/2)/(x/2)]^2, and since sin t/t → 1 as t → 0, the limit is 1/2.