Show lim_{x→∞} sin x / x = 0.

Show lim_{x→∞} sin x / x = 0.

• A. 0
• B. 1
• C. -1
• D. DNE

Answer: (A)

The idea is that a bounded numerator divided by something that grows without bound must shrink toward zero. Since sin x is always between -1 and 1, the absolute value of sin x / x is at most 1/|x|. As x → ∞, 1/|x| → 0, so sin x / x is squeezed to 0. By this reasoning, the limit exists and equals 0. The other possibilities can't occur because the ratio cannot approach 1 or -1 if its magnitude is shrinking to zero, and the bound guarantees a real limit rather than no limit.

The idea is that a bounded numerator divided by something that grows without bound must shrink toward zero. Since sin x is always between -1 and 1, the absolute value of sin x / x is at most 1/|x|. As x → ∞, 1/|x| → 0, so sin x / x is squeezed to 0. By this reasoning, the limit exists and equals 0. The other possibilities can't occur because the ratio cannot approach 1 or -1 if its magnitude is shrinking to zero, and the bound guarantees a real limit rather than no limit.