Which limit equals 1/2?

Which limit equals 1/2?

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

Answer: (D)

This limit relies on how cosine behaves for very small angles. For small x, 1 − cos x is essentially x^2/2. A clean way to see this is the identity 1 − cos x = 2 sin^2(x/2). Then (1 − cos x)/x^2 = [2 sin^2(x/2)] / x^2 = (1/2) [ sin(x/2) / (x/2) ]^2. As x → 0, sin(y)/y → 1 with y = x/2, so the expression approaches (1/2) · 1^2 = 1/2. So the limit equals 1/2. Other common limits in the choices approach different values (for example, arctan x over x tends to 1, and tan x over x tends to 1 near 0, while the ratio x^2/(x^2+1) tends to 1 as x → ∞), so they don’t give 1/2.

This limit relies on how cosine behaves for very small angles. For small x, 1 − cos x is essentially x^2/2. A clean way to see this is the identity 1 − cos x = 2 sin^2(x/2). Then

(1 − cos x)/x^2 = [2 sin^2(x/2)] / x^2 = (1/2) [ sin(x/2) / (x/2) ]^2.

As x → 0, sin(y)/y → 1 with y = x/2, so the expression approaches (1/2) · 1^2 = 1/2.

So the limit equals 1/2. Other common limits in the choices approach different values (for example, arctan x over x tends to 1, and tan x over x tends to 1 near 0, while the ratio x^2/(x^2+1) tends to 1 as x → ∞), so they don’t give 1/2.