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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 4.7.57
Chapter 4, Problem 4.7.57

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


lim_x→π/2⁻ (π/2 - x) sec x

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First, identify the form of the limit as x approaches π/2 from the left. Substitute x = π/2 into the expression (π/2 - x) sec(x) to see if it results in an indeterminate form.
Notice that as x approaches π/2 from the left, (π/2 - x) approaches 0 and sec(x) = 1/cos(x) approaches infinity because cos(x) approaches 0. This results in the indeterminate form 0 * ∞.
To resolve the indeterminate form, rewrite the expression as a fraction: (π/2 - x) sec(x) = (π/2 - x) / cos(x). Now, the limit is in the form 0/0, which is suitable for l'Hôpital's Rule.
Apply l'Hôpital's Rule, which states that if the limit of f(x)/g(x) as x approaches a point is in the form 0/0 or ∞/∞, then it can be evaluated as the limit of f'(x)/g'(x). Differentiate the numerator and the denominator separately.
Differentiate the numerator (π/2 - x) to get -1, and differentiate the denominator cos(x) to get -sin(x). The limit now becomes lim_x→π/2⁻ (-1)/(-sin(x)). Simplify this expression to evaluate the limit as x approaches π/2 from the left.

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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 of discontinuity or 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 ∞/∞. The rule 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, and then re-evaluating the limit. This technique simplifies the process of finding limits in complex scenarios.
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Secant Function

The secant function, denoted as sec(x), is the reciprocal of the cosine function, defined as sec(x) = 1/cos(x). It is important in trigonometry and calculus, particularly when evaluating limits involving trigonometric functions. Understanding the behavior of sec(x) near specific angles, such as π/2, is essential for solving limits that involve this function.
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