Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
f. lim(x→∞) coth x
1
views
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 7.8.3g
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)
Verified step by step guidance1
Identify the function given: \(g(x) = x^{3} e^{-x}\).
Recall that as \(x \to \infty\), the exponential term \(e^{-x}\) approaches 0 very rapidly, while the polynomial term \(x^{3}\) grows without bound but at a much slower rate compared to exponentials.
Analyze the behavior of \(g(x)\) by considering the dominant terms: since \(e^{-x}\) decays faster than any polynomial grows, the product \(x^{3} e^{-x}\) tends to 0 as \(x \to \infty\).
Compare this behavior to \(x^{2}\): since \(g(x)\) tends to 0 and \(x^{2}\) tends to infinity, \(g(x)\) grows slower than \(x^{2}\) as \(x \to \infty\).
Conclude that \(g(x)\) grows slower than \(x^{2}\) as \(x\) approaches infinity.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
3mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Asymptotic Growth Rates
Asymptotic growth rates describe how functions behave as the input grows very large. Comparing growth rates helps determine which functions increase faster, slower, or at the same rate as a reference function, such as x², by analyzing dominant terms and limits at infinity.
Recommended video:
04:16
Intro To Related Rates
Exponential Decay vs Polynomial Growth
Exponential decay functions like e^(-x) decrease rapidly to zero as x approaches infinity, often overpowering polynomial growth terms. When combined, the exponential decay can cause the entire function to approach zero, affecting the overall growth rate compared to pure polynomials.
Recommended video:
09:29Exponential Growth & Decay
Limit Comparison for Growth Rates
To compare growth rates, we use limits of the ratio of two functions as x approaches infinity. If the limit is zero, the numerator grows slower; if infinite, it grows faster; if finite and nonzero, they grow at the same rate. This method rigorously classifies growth behavior.
Recommended video:
04:16Intro To Related Rates
Related Practice
Textbook Question
9. True, or false? As x→∞,
g. ln(x) = o(ln(2x))
1
views
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)
1
views
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