Compute the limit as x approaches 0 of sin(2x) divided by x.

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Multiple Choice

Compute the limit as x approaches 0 of sin(2x) divided by x.

Explanation:
The limit uses the standard fact that sin(t)/t → 1 as t → 0. Here set t = 2x, so sin(2x)/x = [sin(2x)/(2x)] × 2. As x → 0, 2x → 0, and sin(2x)/(2x) → 1, giving a limit of 2. You can also see this with L’Hôpital’s rule: derivative of sin(2x) is 2 cos(2x), derivative of x is 1, so the limit is 2 cos(0) = 2. The result is 2, not 0, 1, or -2.

The limit uses the standard fact that sin(t)/t → 1 as t → 0. Here set t = 2x, so sin(2x)/x = [sin(2x)/(2x)] × 2. As x → 0, 2x → 0, and sin(2x)/(2x) → 1, giving a limit of 2. You can also see this with L’Hôpital’s rule: derivative of sin(2x) is 2 cos(2x), derivative of x is 1, so the limit is 2 cos(0) = 2. The result is 2, not 0, 1, or -2.

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