Which limit equals 1/2?

Which limit equals 1/2?

• A. lim x->0 (1 - cos x)/x^2
• B. lim x->0 (arctan x)/x
• C. lim x->0 (tan x)/x
• D. lim x->∞ (x^2)/(x^2+1)

Answer: (A)

Evaluating this limit leverages a standard small-angle idea and a neat algebraic rewrite. Start with the identity 1 − cos x = 2 sin^2(x/2). Then the expression becomes 2 sin^2(x/2)/x^2. Let u = x/2, so this is 2 sin^2(u)/(4u^2) = (1/2) [sin(u)/u]^2. As x → 0, u → 0, and the well-known limit sin(u)/u → 1. Therefore, the limit equals (1/2)·1^2 = 1/2. For context, the other expressions behave differently near zero or at infinity: arctan x / x → 1, tan x / x → 1, and x^2/(x^2+1) → 1 as x → ∞, so they do not yield 1/2.

Evaluating this limit leverages a standard small-angle idea and a neat algebraic rewrite. Start with the identity 1 − cos x = 2 sin^2(x/2). Then the expression becomes 2 sin^2(x/2)/x^2. Let u = x/2, so this is 2 sin^2(u)/(4u^2) = (1/2) [sin(u)/u]^2. As x → 0, u → 0, and the well-known limit sin(u)/u → 1. Therefore, the limit equals (1/2)·1^2 = 1/2.

For context, the other expressions behave differently near zero or at infinity: arctan x / x → 1, tan x / x → 1, and x^2/(x^2+1) → 1 as x → ∞, so they do not yield 1/2.