What is the limit as x -> 0 of sin x over x?

What is the limit as x -> 0 of sin x over x?

• A. DNE
• B. -1
• C. 1
• D. 0

Answer: (C)

As x gets close to zero, sin x behaves like x, so their ratio sin x / x tends to a stable value. A precise way to see this uses a squeeze: for small positive x, sin x ≤ x, so sin x / x ≤ 1. Also sin x ≥ x cos x, which gives sin x / x ≥ cos x. Thus cos x ≤ sin x / x ≤ 1. As x → 0, cos x → 1, so by the squeeze theorem the limit of sin x / x is 1. Since sin is an odd function, the same limit from the left matches the right, so the overall limit is 1. This is the reason the value is 1.

As x gets close to zero, sin x behaves like x, so their ratio sin x / x tends to a stable value. A precise way to see this uses a squeeze: for small positive x, sin x ≤ x, so sin x / x ≤ 1. Also sin x ≥ x cos x, which gives sin x / x ≥ cos x. Thus cos x ≤ sin x / x ≤ 1. As x → 0, cos x → 1, so by the squeeze theorem the limit of sin x / x is 1. Since sin is an odd function, the same limit from the left matches the right, so the overall limit is 1. This is the reason the value is 1.