What is the limit lim_{x→0} (sin x)/x?

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Multiple Choice

What is the limit lim_{x→0} (sin x)/x?

This limit shows how sine behaves for very small angles. As x approaches zero, sin x behaves like x, so the ratio sin x over x tends to 1. A standard way to see this rigorously is to use the inequality sin x ≤ x ≤ tan x for small positive x (and its symmetric form for negative x). Dividing by x and rearranging gives cos x ≤ sin x / x ≤ 1. Since cos x tends to 1 as x → 0, the squeeze theorem tells us sin x / x must approach 1. Therefore, the limit is 1.

This limit shows how sine behaves for very small angles. As x approaches zero, sin x behaves like x, so the ratio sin x over x tends to 1. A standard way to see this rigorously is to use the inequality sin x ≤ x ≤ tan x for small positive x (and its symmetric form for negative x). Dividing by x and rearranging gives cos x ≤ sin x / x ≤ 1. Since cos x tends to 1 as x → 0, the squeeze theorem tells us sin x / x must approach 1. Therefore, the limit is 1.

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