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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 4.7.77
Chapter 4, Problem 4.7.77

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_Θ→π/2⁻ (tan Θ)ᶜᵒˢ ᶿ

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1
Identify the form of the limit as Θ approaches π/2 from the left. Notice that tan(Θ) approaches infinity and cos(Θ) approaches 0, creating an indeterminate form of the type ∞^0.
To resolve the indeterminate form, take the natural logarithm of the expression. Let y = (tan(Θ))^cos(Θ), then ln(y) = cos(Θ) * ln(tan(Θ)).
Evaluate the limit of ln(y) as Θ approaches π/2 from the left. This becomes lim_Θ→π/2⁻ [cos(Θ) * ln(tan(Θ))].
Recognize that this is an indeterminate form of type 0 * ∞. Rewrite it as a fraction: lim_Θ→π/2⁻ [ln(tan(Θ)) / (1/cos(Θ))].
Apply l'Hôpital's Rule to the limit of ln(y) since it is now in the form ∞/∞. Differentiate the numerator and the denominator separately with respect to Θ, and then evaluate the limit.

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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 concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points where the function may not be explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.
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L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule simplifies the process of finding limits in complex functions.
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Trigonometric Functions

Trigonometric functions, such as tangent (tan) and cosine (cos), are essential in calculus for analyzing periodic phenomena and angles. The tangent function, in particular, can approach infinity as its input approaches certain values, such as π/2. Understanding the behavior of these functions near their critical points is vital for evaluating limits involving trigonometric expressions.
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