Compute lim_{x->0} (sin x)/x.

Compute lim_{x->0} (sin x)/x.

• A. 0
• B. ∞
• C. -1
• D. 1

Answer: (D)

This limit shows how sin x behaves relative to x when x is very close to zero, with x measured in radians. For small positive x, we have sin x < x < tan x, and dividing by sin x gives 1 > sin x / x > cos x. As x approaches 0, cos x tends to 1, so sin x / x is squeezed between quantities that both go to 1. By the squeeze theorem, the limit is 1. Equivalently, it matches the derivative of sin at 0, since d/dx sin x at 0 equals cos 0 = 1.

This limit shows how sin x behaves relative to x when x is very close to zero, with x measured in radians. For small positive x, we have sin x < x < tan x, and dividing by sin x gives 1 > sin x / x > cos x. As x approaches 0, cos x tends to 1, so sin x / x is squeezed between quantities that both go to 1. By the squeeze theorem, the limit is 1. Equivalently, it matches the derivative of sin at 0, since d/dx sin x at 0 equals cos 0 = 1.