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?
e. x^3 - x^2
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.8.5e
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?
e. x
Verified step by step guidance1
Recall that the growth rate of functions as \(x \to \infty\) can be compared using limits of their ratios. Specifically, for two functions \(f(x)\) and \(g(x)\), if \(\lim_{x \to \infty} \frac{f(x)}{g(x)} = \infty\), then \(f(x)\) grows faster than \(g(x)\). If the limit is a finite nonzero constant, they grow at the same rate. If the limit is 0, \(f(x)\) grows slower than \(g(x)\).
Identify the functions to compare: here, \(f(x) = x\) and \(g(x) = \ln(x)\).
Set up the limit to compare their growth rates: \(\lim_{x \to \infty} \frac{x}{\ln(x)}\).
Analyze the limit: since \(x\) grows without bound much faster than \(\ln(x)\), this limit tends to \(\infty\), indicating that \(x\) grows faster than \(\ln(x)\) as \(x \to \infty\).
Conclude that the function \(x\) grows faster than \(\ln(x)\) as \(x\) approaches infinity.
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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 vs. Polynomial Growth
Logarithmic functions like ln(x) grow very slowly compared to polynomial functions such as x. As x approaches infinity, polynomial functions increase much faster than logarithmic ones, making them dominant in growth comparisons.
Recommended video:
07:00Taylor Polynomials
Asymptotic Behavior and Limits
Asymptotic behavior studies the trend of functions as x approaches infinity. Using limits, we can compare the ratio of two functions to determine if one grows faster, slower, or at the same rate as the other.
Recommended video:
5:50Asymptotes of Hyperbolas
Related Practice
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?
e. e^(-x)
Textbook Question
112. True, or false? Give reasons for your answers.
e. sec^(-1)x = O(1)
Textbook Question
1. Which of the following functions grow faster than e^x as x→∞? Which grow at the same rate as e^x? Which grow slower?
e. (3/2)^x
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?
e. x ln(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?
e. x - 2ln(x)
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