What is the value of lim_{x->0} (tan x)/x?

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Multiple Choice

What is the value of lim_{x->0} (tan x)/x?

Explanation:
This question tests how tan x behaves like x as x approaches zero. Since tan x = sin x / cos x, the expression becomes (sin x)/(x cos x) = [(sin x)/x] · [1/cos x]. As x → 0, the standard small-angle limits give sin x / x → 1 and cos x → 1, so 1/cos x → 1 as well. The product of these two factors tends to 1, so the limit of tan x over x is 1.

This question tests how tan x behaves like x as x approaches zero. Since tan x = sin x / cos x, the expression becomes (sin x)/(x cos x) = [(sin x)/x] · [1/cos x]. As x → 0, the standard small-angle limits give sin x / x → 1 and cos x → 1, so 1/cos x → 1 as well. The product of these two factors tends to 1, so the limit of tan x over x is 1.

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