Use l’Hôpital’s rule to find the limits in Exercises 7–52.

19. lim (θ → π/6) (sin θ - 1/2) / (θ - π/6)

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Identify the limit expression: \(\lim_{\theta \to \frac{\pi}{6}} \frac{\sin \theta - \frac{1}{2}}{\theta - \frac{\pi}{6}}\).

Check if the limit is an indeterminate form by substituting \(\theta = \frac{\pi}{6}\): since \(\sin \frac{\pi}{6} = \frac{1}{2}\), the numerator becomes \(\frac{1}{2} - \frac{1}{2} = 0\) and the denominator is \(\frac{\pi}{6} - \frac{\pi}{6} = 0\), so the limit is of the form \(\frac{0}{0}\).

Since the limit is an indeterminate form \(\frac{0}{0}\), apply l’Hôpital’s Rule by differentiating the numerator and denominator separately with respect to \(\theta\).

Compute the derivative of the numerator: \(\frac{d}{d\theta} (\sin \theta - \frac{1}{2}) = \cos \theta\), and the derivative of the denominator: \(\frac{d}{d\theta} (\theta - \frac{\pi}{6}) = 1\).

Rewrite the limit using these derivatives: \(\lim_{\theta \to \frac{\pi}{6}} \frac{\cos \theta}{1}\), then evaluate this limit by substituting \(\theta = \frac{\pi}{6}\).

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주요 개념

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Limits and Indeterminate Forms

Limits describe the behavior of a function as the input approaches a particular value. When direct substitution results in an indeterminate form like 0/0, special techniques such as l’Hôpital’s rule are needed to evaluate the limit.

l’Hôpital’s Rule

l’Hôpital’s rule states that if a limit yields an indeterminate form 0/0 or ∞/∞, the limit of the ratio of the functions equals the limit of the ratio of their derivatives, provided this latter limit exists. It simplifies evaluating tricky limits.

Derivative of Trigonometric Functions

Understanding the derivatives of sine and cosine functions is essential when applying l’Hôpital’s rule to trigonometric limits. For example, the derivative of sin(θ) is cos(θ), which is used to find the new limit after differentiation.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.19. lim (θ → π/6) (sin θ - 1/2) / (θ - π/6)