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

17. lim (θ → π/2) (2θ - π) / cos(2π - θ)

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First, identify the form of the limit as \( \theta \to \frac{\pi}{2} \). Substitute \( \theta = \frac{\pi}{2} \) into the expression \( \frac{2\theta - \pi}{\cos(2\pi - \theta)} \) to check if it results in an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

Since direct substitution gives \( 2\left(\frac{\pi}{2}\right) - \pi = 0 \) and \( \cos(2\pi - \frac{\pi}{2}) = \cos\left(\frac{3\pi}{2}\right) = 0 \), the limit is of the indeterminate form \( \frac{0}{0} \), so l’Hôpital’s Rule applies.

Apply l’Hôpital’s Rule by differentiating the numerator and denominator separately with respect to \( \theta \). The derivative of the numerator \( 2\theta - \pi \) is \( 2 \).

The derivative of the denominator \( \cos(2\pi - \theta) \) requires the chain rule: \( \frac{d}{d\theta} \cos(2\pi - \theta) = -\sin(2\pi - \theta) \times (-1) = \sin(2\pi - \theta) \).

Rewrite the limit as \( \lim_{\theta \to \frac{\pi}{2}} \frac{2}{\sin(2\pi - \theta)} \) and then substitute \( \theta = \frac{\pi}{2} \) to evaluate the limit.

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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 certain value. When direct substitution results in forms like 0/0 or ∞/∞, these are called indeterminate forms, which require special techniques such as l’Hôpital’s rule to evaluate.

l’Hôpital’s Rule

l’Hôpital’s rule is a method for evaluating limits that yield indeterminate forms 0/0 or ∞/∞ by differentiating the numerator and denominator separately and then taking the limit of their quotient.

Trigonometric Functions and Their Limits

Understanding the behavior of trigonometric functions like cosine near specific points is essential. Knowing how to simplify or differentiate these functions helps in applying l’Hôpital’s rule effectively to find limits.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.17. lim (θ → π/2) (2θ - π) / cos(2π - θ)