Using L'Hôpital's rule, what is lim x->0 (sin x)/x?

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Multiple Choice

Using L'Hôpital's rule, what is lim x->0 (sin x)/x?

Explanation:
This limit hinges on a quotient where both numerator and denominator vanish as x approaches 0, so L'Hôpital's rule applies. Take the derivatives of top and bottom: the derivative of sin x is cos x, and the derivative of x is 1. The limit becomes lim x->0 cos x / 1, which evaluates to cos(0) = 1. So the limit is 1. Intuitively, near zero sin x behaves like x, so their ratio approaches 1. The result isn’t 0 or infinity, and it isn’t -1, because the exact evaluation with the derivatives gives 1.

This limit hinges on a quotient where both numerator and denominator vanish as x approaches 0, so L'Hôpital's rule applies. Take the derivatives of top and bottom: the derivative of sin x is cos x, and the derivative of x is 1. The limit becomes lim x->0 cos x / 1, which evaluates to cos(0) = 1. So the limit is 1.

Intuitively, near zero sin x behaves like x, so their ratio approaches 1. The result isn’t 0 or infinity, and it isn’t -1, because the exact evaluation with the derivatives gives 1.

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