Evaluate each limit.

lim θ→0 (1/(2+sinθ)-1/2)/sin θ

검증된 단계별 안내

Recognize that the given limit is in the indeterminate form 0/0 as \( \theta \to 0 \). This suggests the use of L'Hôpital's Rule, which is applicable for limits of the form 0/0 or \( \infty/\infty \).

Apply L'Hôpital's Rule, which involves differentiating the numerator and the denominator separately. The original expression is \( \frac{\frac{1}{2+\sin\theta} - \frac{1}{2}}{\sin\theta} \).

Differentiate the numerator: The derivative of \( \frac{1}{2+\sin\theta} \) with respect to \( \theta \) is \( -\frac{\cos\theta}{(2+\sin\theta)^2} \). The derivative of \( \frac{1}{2} \) is 0.

Differentiate the denominator: The derivative of \( \sin\theta \) with respect to \( \theta \) is \( \cos\theta \).

Substitute the derivatives back into the limit expression: \( \lim_{\theta \to 0} \frac{-\frac{\cos\theta}{(2+\sin\theta)^2}}{\cos\theta} \). Simplify the expression and evaluate the limit as \( \theta \to 0 \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of a function as θ approaches 0. Understanding limits is crucial for evaluating expressions that may be indeterminate or undefined at specific points.

Trigonometric Functions

Trigonometric functions, such as sine (sin), are essential in calculus, especially when dealing with angles and periodic phenomena. The sine function is particularly important in this limit problem, as it approaches 0 when θ approaches 0. Familiarity with the properties and behavior of trigonometric functions helps in simplifying and evaluating limits involving these functions.

L'Hôpital's Rule

L'Hôpital's Rule is a technique used to evaluate limits that result in indeterminate forms like 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to such forms, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule can be particularly useful in solving the limit presented in the question.

Evaluate each limit. lim θ→0 (1/(2+sinθ)-1/2)/sin θ