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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 4.7.72
Chapter 4, Problem 4.7.72

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→π/2⁻ (π - 2x) tan 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 - 2x) tan x to check if it results in an indeterminate form like 0/0 or ∞/∞.
Substitute x = π/2 into the expression: (π - 2x) becomes 0, and tan x approaches ∞ as x approaches π/2 from the left. This results in the indeterminate form 0 * ∞, which can be rewritten as 0/0 by expressing tan x as sin x / cos x.
Rewrite the expression as (π - 2x) / (1/tan x) = (π - 2x) / (cos x/sin x) = (π - 2x) sin x / cos x. This is now in the form 0/0, which allows us to apply 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 results in an indeterminate form, then the limit is the same as the limit of f'(x)/g'(x) as x approaches that point. Differentiate the numerator and the denominator separately.
Differentiate the numerator (π - 2x) to get -2, and differentiate the denominator sin x / cos x using the quotient rule. The derivative of sin x is cos x, and the derivative of cos x is -sin x. Apply these derivatives to find the new limit.

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Key Concepts

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

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's behavior near points of interest, including points of discontinuity or where the function is not explicitly defined. Evaluating limits is essential for determining the continuity and differentiability of functions.
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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. This process can be repeated if the result remains indeterminate.
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Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental in calculus, particularly when dealing with limits involving angles. The behavior of these functions near specific points, like π/2, can lead to unique limit scenarios, such as vertical asymptotes. Understanding their properties and graphs is crucial for evaluating limits that involve trigonometric expressions.
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Introduction to Trigonometric Functions
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