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 7.8.5g
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
Verified step by step guidance1
Recall that as \(x \to \infty\), the natural logarithm function \(\ln(x)\) grows without bound, but very slowly compared to many other functions.
Analyze the function \(g(x) = \frac{1}{x}\). As \(x \to \infty\), \(\frac{1}{x} \to 0\), which means it approaches zero rather than increasing.
Since \(\ln(x) \to \infty\) and \(\frac{1}{x} \to 0\), \(\frac{1}{x}\) grows slower than \(\ln(x)\) as \(x\) becomes very large.
To compare growth rates formally, consider the limit \(\lim_{x \to \infty} \frac{g(x)}{\ln(x)} = \lim_{x \to \infty} \frac{1/x}{\ln(x)} = \lim_{x \to \infty} \frac{1}{x \ln(x)}\).
Since \(x \ln(x) \to \infty\), the limit above is \(0\), confirming that \(g(x) = \frac{1}{x}\) grows slower than \(\ln(x)\) 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 as x Approaches Infinity
This concept involves comparing how quickly different functions increase as the input variable x becomes very large. Understanding growth rates helps classify functions as growing faster, slower, or at the same rate relative to each other, often using limits or asymptotic behavior.
Recommended video:
05:36
Integrals of Natural Exponential Functions (e^x) Example 3
Behavior of the Natural Logarithm Function ln(x)
The natural logarithm function ln(x) increases without bound as x approaches infinity, but it does so very slowly compared to polynomial or exponential functions. Recognizing its slow growth rate is key to comparing it with other functions.
Recommended video:
05:18Derivative of the Natural Logarithmic Function
Limits and Dominance in Comparing Functions
Using limits, especially the limit of the ratio of two functions as x approaches infinity, helps determine which function grows faster. If the limit is zero, the numerator grows slower; if infinite, it grows faster; if finite and nonzero, they grow at the same rate.
Recommended video:
06:11Limits of Rational Functions: Denominator = 0
Related Practice
Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
f. lim(x→∞) coth x
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Textbook Question
9. True, or false? As x→∞,
g. ln(x) = o(ln(2x))
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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?
f. (e^x)/2
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))