Textbook Question
In Exercises 1–6, use l’Hôpital’s Rule to evaluate the limit. Then evaluate the limit using a method studied in Chapter 2.
5. lim (x → 0) (1 - cos x) / x²
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.5.12
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
12. lim (x → ∞) (x - 8x²) / (12x² + 5x)
Verified step by step guidance1
Identify the limit expression: \(\lim_{x \to \infty} \frac{x - 8x^{2}}{12x^{2} + 5x}\).
Check the form of the limit by analyzing the degrees of the numerator and denominator as \(x\) approaches infinity. Both numerator and denominator tend to infinity, so the limit is of the form \(\frac{\infty}{\infty}\), which is an indeterminate form suitable for l'Hôpital's Rule.
Apply l'Hôpital's Rule by differentiating the numerator and denominator separately with respect to \(x\): differentiate numerator \(\frac{d}{dx}(x - 8x^{2})\) and denominator \(\frac{d}{dx}(12x^{2} + 5x)\).
Write the new limit expression using the derivatives: \(\lim_{x \to \infty} \frac{\frac{d}{dx}(x - 8x^{2})}{\frac{d}{dx}(12x^{2} + 5x)}\).
Evaluate the new limit by simplifying the derivatives and then analyzing the behavior as \(x\) approaches infinity to find the limit.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits at Infinity
Limits at infinity describe the behavior of a function as the input grows without bound. Understanding how to evaluate these limits helps determine the end behavior of rational functions, often by comparing the degrees of polynomials in the numerator and denominator.
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Cases Where Limits Do Not Exist
Indeterminate Forms
Indeterminate forms like ∞/∞ or 0/0 occur when direct substitution in a limit does not yield a clear answer. Recognizing these forms is essential because they signal the need for techniques like l’Hôpital’s rule to evaluate the limit properly.
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l’Hôpital’s Rule
l’Hôpital’s rule provides a method to evaluate limits that result in indeterminate forms by differentiating the numerator and denominator separately. Applying this rule simplifies the limit calculation, especially for rational functions where direct substitution is inconclusive.
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Related Practice
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
49. lim (x → 0) (x - sin x) / (x tan x)
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Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
33. y=ln(arctan(x))
Textbook Question
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
61. y = √(θ + 3) sin θ
Textbook Question
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
64. y = 1/(t(t+1)(t+2))
Textbook Question
Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.
f(x)=x³-1