Which limit equals 2?

Which limit equals 2?

• A. lim_{x->0} sin x / x
• B. lim_{x->0} tan x / x
• C. lim_{x->0} sin(2x)/x
• D. lim_{x->0} (1 - cos x)/x^2

Answer: (C)

Small-angle behavior of sine and cosine is the key here: near zero, sin t behaves like t, tan t behaves like t, and cos t can be approximated by 1 − t^2/2. A fundamental tool is the limit lim as u→0 of sin u over u equals 1. For the expression sin(2x) over x, use a substitution u = 2x. Then sin(2x)/x = [sin(2x)/(2x)] × 2. As x → 0, u → 0, so sin(u)/u → 1, and the product tends to 1 × 2 = 2. Therefore this limit is 2. The other limits approach different values: sin x over x tends to 1, tan x over x tends to 1, and (1 − cos x)/x^2 tends to 1/2. Hence the expression sin(2x)/x is the one that equals 2.

Small-angle behavior of sine and cosine is the key here: near zero, sin t behaves like t, tan t behaves like t, and cos t can be approximated by 1 − t^2/2. A fundamental tool is the limit lim as u→0 of sin u over u equals 1.

For the expression sin(2x) over x, use a substitution u = 2x. Then sin(2x)/x = [sin(2x)/(2x)] × 2. As x → 0, u → 0, so sin(u)/u → 1, and the product tends to 1 × 2 = 2. Therefore this limit is 2.

The other limits approach different values: sin x over x tends to 1, tan x over x tends to 1, and (1 − cos x)/x^2 tends to 1/2. Hence the expression sin(2x)/x is the one that equals 2.