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

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

• A. 0
• B. 1
• C. 2
• D. 4

Answer: (C)

For small x, cos(2x) behaves like 1 minus (2x) squared over 2, so 1 - cos(2x) is about 2x^2. Dividing by x^2 gives a limit of 2. A clean way to see this exactly is to use the identity 1 - cos(2x) = 2 sin^2(x). Then the expression becomes 2 (sin x / x)^2, and since sin x / x → 1 as x → 0, the limit is 2.

For small x, cos(2x) behaves like 1 minus (2x) squared over 2, so 1 - cos(2x) is about 2x^2. Dividing by x^2 gives a limit of 2. A clean way to see this exactly is to use the identity 1 - cos(2x) = 2 sin^2(x). Then the expression becomes 2 (sin x / x)^2, and since sin x / x → 1 as x → 0, the limit is 2.