Textbook Question
86. This exercise explores the difference between
lim(x→∞)(1 + 1/x²)^x
and
lim(x→∞)(1 + 1/x)^x = e
c. Confirm your estimate of lim(x→∞)f(x) by calculating it with l’Hôpital’s Rule.
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 7.8.5c
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?
c. ln(√x)
Verified step by step guidance1
Recall that the growth rate of functions as \(x \to \infty\) can be compared by analyzing their dominant terms or by using limits of their ratios.
Rewrite the given function \(\ln(\sqrt{x})\) using logarithm properties: \(\ln(\sqrt{x}) = \ln(x^{1/2}) = \frac{1}{2} \ln(x)\).
Compare \(\ln(\sqrt{x})\) to \(\ln(x)\) by considering the ratio \(\frac{\ln(\sqrt{x})}{\ln(x)} = \frac{\frac{1}{2} \ln(x)}{\ln(x)} = \frac{1}{2}\).
Since the ratio approaches a positive constant (\(\frac{1}{2}\)), \(\ln(\sqrt{x})\) grows at the same rate as \(\ln(x)\), but scaled by a constant factor.
Therefore, \(\ln(\sqrt{x})\) neither grows faster nor slower than \(\ln(x)\) in terms of growth rate classification; they grow at the same rate.
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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 as x Approaches Infinity
Understanding how functions behave as x approaches infinity is essential for comparing their growth rates. Functions can grow faster, slower, or at the same rate depending on their dominant terms. This concept helps classify functions by their long-term behavior, such as logarithmic, polynomial, or exponential growth.
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Integrals of Natural Exponential Functions (e^x) Example 3
Properties of Logarithmic Functions
Logarithmic functions like ln(x) grow slowly compared to polynomials and exponentials. Key properties include the logarithm of a root, ln(√x) = (1/2)ln(x), which shows that scaling inside the log translates to a constant multiple outside. This helps compare growth rates by simplifying expressions.
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06:21Properties of Functions
Asymptotic Equivalence and Big-O Notation
Asymptotic equivalence means two functions grow at the same rate if their ratio approaches a nonzero constant as x→∞. Big-O notation formalizes this by describing upper bounds on growth. Recognizing when functions differ by constant multiples is crucial for determining if they grow at the same rate.
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03:38Intro to Continuity Example 1
Related Practice
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
7. c. arcsec(-2)
Textbook Question
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
1. 2y' + 3y = e^(-x)
c. y = e^(-x) + Ce^(-(3/2)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?
c. 1/√x
Textbook Question
80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.
c. f(x) = x³/ (3 - 4x), g(x) = x², (a, b) = (0, 3)
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?
c. √(1+x^4)
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