Textbook Question
22. The function ln x grows slower than any polynomial Show that ln(x) grows slower as x→∞ than any nonconstant polynomial.
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 7.6.15
Find the limits in Exercises 13–20. (If in doubt, look at the function’s graph.)
15. lim(x→∞)arctan(x)
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Recall the definition and behavior of the arctangent function, \(\arctan(x)\), which is the inverse of the tangent function restricted to \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\).
Understand that as \(x\) approaches infinity, the value of \(\arctan(x)\) approaches a horizontal asymptote because the tangent function has vertical asymptotes at \(\pm \frac{\pi}{2}\).
Recognize that \(\arctan(x)\) is an increasing function and its range is \(\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\), so the limit as \(x \to \infty\) must be the upper bound of this range.
Express the limit formally as \(\lim_{x \to \infty} \arctan(x) = L\), where \(L\) is the horizontal asymptote value to be identified.
Conclude that the limit is the horizontal asymptote \(L = \frac{\pi}{2}\), since \(\arctan(x)\) approaches this value as \(x\) becomes very large.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limit of a Function at Infinity
The limit of a function as x approaches infinity describes the value that the function approaches as x grows larger without bound. It helps determine the end behavior of the function, which is essential for understanding asymptotic behavior and horizontal asymptotes.
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Limits of Rational Functions: Denominator = 0
Arctangent Function Properties
The arctangent function, denoted arctan(x), is the inverse of the tangent function restricted to its principal domain. It is continuous and increasing, with horizontal asymptotes at ±π/2, meaning as x approaches infinity, arctan(x) approaches π/2.
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06:21Properties of Functions
Using Graphs to Understand Limits
Graphs provide a visual representation of a function’s behavior, making it easier to estimate limits, especially at infinity. Observing the graph of arctan(x) shows how the function approaches its horizontal asymptote, reinforcing the analytical limit result.
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Related Practice
Textbook Question
Each of Exercises 25–36 gives a formula for a function y=f(x). In each case, find f^(-1)(x) and identify the domain and range of f^(-1). As a check, show that f(f^(-1)(x))=f^(-1)(f(x))=x.
f(x) = x⁵
Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
119. y = (sin x)ˣ
Textbook Question
147. Find the area of the region between the curve y = 2x / (1 + x²) and the interval −2 ≤ x ≤ 2 of the x-axis.
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Textbook Question
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
3. y = 1/x ∫(from 1 to x) e^t/t dt, x²y' + xy = e^x
Textbook Question
80. Volume The region enclosed by the curve y=sech(x), the x-axis, and the lines x=±ln√3 is revolved about the x-axis to generate a solid. Find the volume of the solid.
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