Determine the following limits.lim⁡θ→0−sin⁡θcos⁡2θ−1{\displaystyle\lim_{\theta\$$to0^{-}$$}\frac{\sin\theta}{\$$cos^2$$\theta-1}}

Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 2 - Limits

문제 2.4.44

2장, 문제 2.4.44

Determine the following limits.

$\lim_{θ \to 0^{-}} \frac{\sin θ}{\cos^{2} θ - 1}$

검증된 단계별 안내

Step 1: Recognize that the expression $ \cos^2\theta - 1 $ can be rewritten using the Pythagorean identity $ \cos^2\theta = 1 - \sin^2\theta $. Therefore, $ \cos^2\theta - 1 = -(\sin^2\theta) $.

Step 2: Substitute $ \cos^2\theta - 1 $ with $ -\sin^2\theta $ in the limit expression. The limit now becomes $ \lim_{\theta \to 0^{-}} \frac{\sin\theta}{-\sin^2\theta} $.

Step 3: Simplify the expression by canceling one $ \sin\theta $ from the numerator and the denominator. This results in $ \lim_{\theta \to 0^{-}} \frac{1}{-\sin\theta} $.

Step 4: Evaluate the limit as $ \theta \to 0^{-} $. As $ \theta $ approaches 0 from the negative side, $ \sin\theta $ approaches 0 from the negative side, making $ \frac{1}{-\sin\theta} $ approach negative infinity.

Step 5: Conclude that the limit is $-\infty$ as $ \theta \to 0^{-} $.

비슷한 문제에 대한 검증된 영상 답변:

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주요 개념

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

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this case, we are examining the limit of a function as theta approaches 0 from the left (0-). Understanding limits is crucial for analyzing the behavior of functions near points of interest, especially when direct substitution may lead to indeterminate forms.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In this limit problem, the sine function appears in the numerator, while the cosine function is involved in the denominator. Familiarity with the properties and behaviors of these functions, particularly near key angles like 0, is essential for evaluating the limit accurately.

Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In this problem, substituting theta = 0 results in the form 0/0, necessitating further analysis, such as algebraic manipulation or L'Hôpital's Rule, to resolve the limit and find the correct value.

Determine the following limits.limθ→0−sinθcos2θ−1{\(\displaystyle\[\lim\)_{\(\theta\]\to\)0^{-}}\(\frac{\sin\theta}{\cos^2\theta-1}\)}