Textbook Question
3. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
g. x^3 e^(-x)
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 7.8.9.g
9. True, or false? As x→∞,
g. ln(x) = o(ln(2x))
Verified step by step guidance1
Recall the definition of the little-o notation: for functions f(x) and g(x), we say f(x) = o(g(x)) as x → ∞ if \( \lim_{x \to \infty} \frac{f(x)}{g(x)} = 0 \).
Identify the functions in the problem: \( f(x) = \ln(x) \) and \( g(x) = \ln(2x) \). We need to analyze \( \lim_{x \to \infty} \frac{\ln(x)}{\ln(2x)} \).
Use logarithm properties to simplify the denominator: \( \ln(2x) = \ln(2) + \ln(x) \). So the limit becomes \( \lim_{x \to \infty} \frac{\ln(x)}{\ln(2) + \ln(x)} \).
Divide numerator and denominator by \( \ln(x) \) to simplify the expression: \( \lim_{x \to \infty} \frac{1}{\frac{\ln(2)}{\ln(x)} + 1} \).
Analyze the behavior as \( x \to \infty \): since \( \ln(x) \to \infty \), \( \frac{\ln(2)}{\ln(x)} \to 0 \), so the limit approaches \( \frac{1}{0 + 1} = 1 \). Since the limit is 1, not 0, the statement \( \ln(x) = o(\ln(2x)) \) is false.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Little-o Notation (Asymptotic Comparison)
Little-o notation, written as f(x) = o(g(x)), means that f(x) grows much slower than g(x) as x approaches infinity, specifically that the limit of f(x)/g(x) is zero. It is used to compare the relative growth rates of functions.
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Intro to Continuity Example 1
Properties of Logarithmic Functions
Logarithmic functions like ln(x) grow without bound but very slowly as x approaches infinity. Key properties include ln(ab) = ln(a) + ln(b), which helps simplify expressions like ln(2x) into ln(2) + ln(x).
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06:21Properties of Functions
Limits Involving Logarithmic Functions at Infinity
When evaluating limits of ratios of logarithmic functions as x→∞, constants become negligible. For example, ln(2x) = ln(2) + ln(x), so the ratio ln(x)/ln(2x) approaches 1, indicating similar growth rates.
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5:26Graphs of Logarithmic Functions
Related Practice
Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
h. lim(x→0^-) coth x
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Textbook Question
5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
g. 1/x
Textbook Question
6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
g. ln(ln x)
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Textbook Question
2. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
g. e^(cos(x))
Textbook Question
4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
g. (1.1)^x