Textbook Question
112. True, or false? Give reasons for your answers.
a. 1/x⁴ = O(1/x² + 1/x⁴)
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.P.111c
111. True, or false? Give reasons for your answers.
c. x = o(x + ln(x))
Verified step by step guidance1
Understand the notation: The expression \(x = o(x + \ln(x))\) means that \(x\) is a little-o of \(x + \ln(x)\) as \(x\) approaches some limit (usually \(\infty\) or 0). This implies that \(\lim_{x \to a} \frac{x}{x + \ln(x)} = 0\), where \(a\) is the point of interest.
Determine the limit point: Since \(\ln(x)\) is defined for \(x > 0\), the natural domain to consider is \(x \to \infty\) or \(x \to 0^+\). We will analyze the limit \(\lim_{x \to \infty} \frac{x}{x + \ln(x)}\).
Set up the limit: Write the ratio explicitly as \(\lim_{x \to \infty} \frac{x}{x + \ln(x)}\) and consider the behavior of numerator and denominator as \(x\) grows large.
Simplify the expression inside the limit by dividing numerator and denominator by \(x\): \(\lim_{x \to \infty} \frac{1}{1 + \frac{\ln(x)}{x}}\).
Evaluate the limit: Since \(\frac{\ln(x)}{x} \to 0\) as \(x \to \infty\), the limit becomes \(\frac{1}{1 + 0} = 1\), which is not zero. Therefore, \(x\) is not \(o(x + \ln(x))\) as \(x \to \infty\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Little-o Notation (o-notation)
Little-o notation describes the limiting behavior of a function relative to another as the input approaches a point, usually zero or infinity. Specifically, f(x) = o(g(x)) means that f(x) grows much slower than g(x), or equivalently, the limit of f(x)/g(x) as x approaches the point is zero.
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Sigma Notation
Behavior of the Natural Logarithm Near Zero
The natural logarithm function ln(x) approaches negative infinity as x approaches zero from the right. This unbounded behavior affects limits and comparisons involving ln(x) near zero, making expressions like x + ln(x) behave differently than simple polynomial terms.
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Limit Comparison for Composite Functions
To determine if an expression like x = o(x + ln(x)) holds, one must analyze the limit of the ratio x / (x + ln(x)) as x approaches zero. Understanding how to compare growth rates and evaluate such limits is essential to verify or refute statements involving little-o notation.
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Related Practice
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
13. y = (x+2)^(x+2)
Textbook Question
112. True, or false? Give reasons for your answers.
c. ln x = o(x+1)
1
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Textbook Question
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
29. y = (sin θ)^√θ
Textbook Question
118. A particle is traveling upward and to the right along the curve y=ln(x). Its x-coordinate is increasing at the rate (dx/dt)=√x m/sec. At what rate is the y-coordinate changing at the point (e², 2)?
Textbook Question
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
25. y = 2(x² + 1)/√(cos 2x)