Textbook Question
Evaluate the integrals in Exercises 31–78.
39. ∫(from 0 to π)tan(x/3)dx
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 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.PE.109b
109. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
b. f(x)=x, g(x)=x + 1/x
Verified step by step guidance1
Identify the given functions: \(f(x) = x\) and \(g(x) = x + \frac{1}{x}\).
As \(x \to \infty\), analyze the behavior of each function. For \(f(x) = x\), the function grows without bound linearly.
For \(g(x) = x + \frac{1}{x}\), note that \(\frac{1}{x} \to 0\) as \(x \to \infty\), so \(g(x)\) behaves like \(x\) plus a very small positive term.
Compare the growth rates by considering the ratio \(\frac{f(x)}{g(x)} = \frac{x}{x + \frac{1}{x}}\). Simplify this expression to understand the limit as \(x \to \infty\).
Evaluate the limit \(\lim_{x \to \infty} \frac{f(x)}{g(x)}\) to determine if it approaches 0, a finite nonzero constant, or infinity, which tells us if \(f\) grows slower, at the same rate, or faster than \(g\) respectively.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Asymptotic Behavior of Functions
Asymptotic behavior describes how functions behave as the input grows very large (x→∞). It helps compare growth rates by focusing on dominant terms and ignoring smaller ones that become insignificant at infinity.
Recommended video:
5:50
Asymptotes of Hyperbolas
Limits and Limit Comparison
Limits evaluate the value a function approaches as x approaches infinity. Comparing the limit of the ratio f(x)/g(x) as x→∞ determines if one function grows faster, slower, or at the same rate as the other.
Recommended video:
07:45Limit Comparison Test
Dominant Terms in Functions
The dominant term in a function is the part that grows fastest as x→∞. For example, in g(x) = x + 1/x, the term x dominates because 1/x approaches zero, so the growth rate is mainly determined by x.
Recommended video:
05:44Divergence Test (nth Term Test)
Related Practice
Textbook Question
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
29. y = (sin θ)^√θ
Textbook Question
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
99. lim(x→0) (2^sin(x) - 1)/(e^x - 1)
Textbook Question
Evaluate the integrals in Exercises 31–78.
67. ∫(from -2 to 2)3dt/(4+3t²)
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
7. y = log₂(x²/2)
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
21. y = z arcsec(z) - √(z² - 1), z>1