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 7.8.6g
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)
Verified step by step guidance1
Recall that when comparing growth rates of functions as \(x \to \infty\), we often use limits of their ratios to determine which grows faster, slower, or at the same rate.
Identify the two functions to compare: \(f(x) = \ln(x)\) and \(g(x) = \ln(\ln x)\), where \(x\) is large enough so that \(\ln x > 0\).
Consider the limit \(\lim_{x \to \infty} \frac{g(x)}{f(x)} = \lim_{x \to \infty} \frac{\ln(\ln x)}{\ln x}\). This limit will help us understand their relative growth rates.
Analyze the behavior of the limit: since \(\ln x\) grows without bound but more slowly than any power of \(x\), and \(\ln(\ln x)\) grows even more slowly, the numerator grows much slower than the denominator.
Conclude that because the limit tends to zero, \(g(x) = \ln(\ln x)\) grows slower than \(f(x) = \ln x\) 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.
Growth Rates of Functions
Growth rates describe how functions behave as their input becomes very large. Comparing growth rates helps determine which functions increase faster, slower, or at the same pace as others, especially as x approaches infinity.
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04:16
Intro To Related Rates
Logarithmic Functions and Their Properties
Logarithmic functions like ln(x) grow slowly compared to polynomial or exponential functions. Understanding properties of ln(x) and nested logarithms such as ln(ln x) is essential to compare their growth rates accurately.
Recommended video:
06:21Properties of Functions
Asymptotic Comparison Using Limits
To compare growth rates, limits of ratios of functions as x approaches infinity are used. If the limit of f(x)/g(x) is zero, f grows slower; if infinite, f grows faster; if finite and nonzero, they grow at the same rate.
Recommended video:
07:45Limit Comparison Test
Related Practice
Textbook Question
9. True, or false? As x→∞,
g. ln(x) = o(ln(2x))
1
views
Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
h. lim(x→0^-) coth x
1
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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
82. Use the definitions of the hyperbolic functions to find each of the following limits.
i. lim(x→-∞) csch 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