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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 4.7.30
Chapter 4, Problem 4.7.30

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


lim_x→2π (x sin x + x² - 4π²) / (x - 2π)

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1
Identify the form of the limit as x approaches 2π. Substitute x = 2π into the expression to check if it results in an indeterminate form like 0/0 or ∞/∞.
Substitute x = 2π into the numerator: x sin x + x² - 4π². This results in 2π sin(2π) + (2π)² - 4π², which simplifies to 0.
Substitute x = 2π into the denominator: x - 2π, which also results in 0. Therefore, the limit is in the indeterminate form 0/0, making l'Hôpital's Rule applicable.
Apply l'Hôpital's Rule, which involves differentiating the numerator and the denominator separately. Differentiate the numerator: d/dx(x sin x + x² - 4π²) = sin x + x cos x + 2x.
Differentiate the denominator: d/dx(x - 2π) = 1. Now, evaluate the limit of the new expression (sin x + x cos x + 2x) / 1 as x approaches 2π.

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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 ∞/∞. 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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Continuity

Continuity refers to a property of functions where they do not have any abrupt changes, jumps, or breaks at a given point. A function is continuous at a point if the limit as the input approaches that point equals the function's value at that point. Understanding continuity is essential for evaluating limits and applying L'Hôpital's Rule effectively.
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Related Practice
Textbook Question

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.


ƒ(x) = x³ -4a²x

Textbook Question

Particular antiderivatives For the following functions f, find the antiderivative F that satisfies the given condition.

f(x) = (3y + 5)/y; F(1) = 3. y > 0

Textbook Question

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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Textbook Question

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = -2x⁴ + x² + 10

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Textbook Question

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.


ƒ(x) = 3x⁵ - 25x³ + 60x on [-2,3]

Textbook Question

Rectangles beneath a semicircle A rectangle is constructed with its base on the diameter of a semicircle with radius 5 and its two other vertices on the semicircle. What are the dimensions of the rectangle with maximum area?