Use l’Hôpital’s rule to find the limits in Exercises 7–52.
51. lim (θ → 0) (θ - sin θ cos θ) / (tan θ - θ)

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First, identify the form of the limit as \( \theta \to 0 \). Substitute \( \theta = 0 \) into the numerator and denominator separately to check if the limit is an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \).

Since direct substitution gives an indeterminate form \( \frac{0}{0} \), apply l'Hôpital's Rule, which states that \( \lim_{\theta \to 0} \frac{f(\theta)}{g(\theta)} = \lim_{\theta \to 0} \frac{f'(\theta)}{g'(\theta)} \) provided the latter limit exists.

Compute the derivative of the numerator: \( f(\theta) = \theta - \sin \theta \cos \theta \). Use the product rule for \( \sin \theta \cos \theta \) and the derivative of \( \theta \).

Compute the derivative of the denominator: \( g(\theta) = \tan \theta - \theta \). Use the derivative of \( \tan \theta \) and the derivative of \( \theta \).

After finding \( f'(\theta) \) and \( g'(\theta) \), write the new limit as \( \lim_{\theta \to 0} \frac{f'(\theta)}{g'(\theta)} \). Then, substitute \( \theta = 0 \) into this expression to evaluate the limit.

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주요 개념

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l’Hôpital’s Rule

l’Hôpital’s Rule is a method for evaluating limits that result in indeterminate forms like 0/0 or ∞/∞. It states that the limit of a ratio of functions can be found by taking the limit of the ratio of their derivatives, provided certain conditions are met. This rule simplifies complex limit problems involving trigonometric or other functions.

Trigonometric Limits and Identities

Understanding basic trigonometric limits, such as lim θ→0 (sin θ)/θ = 1 and lim θ→0 (tan θ)/θ = 1, is essential. Familiarity with trigonometric identities helps simplify expressions before or after applying l’Hôpital’s Rule, making it easier to evaluate the limit accurately.

Derivative of Trigonometric Functions

Knowing how to differentiate trigonometric functions like sin θ, cos θ, and tan θ is crucial when applying l’Hôpital’s Rule. For example, the derivatives are cos θ, -sin θ, and sec² θ respectively. Correct differentiation allows the limit to be evaluated correctly after applying the rule.

Use l’Hôpital’s rule to find the limits in Exercises 7–52.51. lim (θ → 0) (θ - sin θ cos θ) / (tan θ - θ)