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

Problem 2.4.44

Chapter 2, Problem 2.4.44

Determine the following limits.

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

Verified step by step guidance

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^{-} $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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