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?
a. log_2(x²)
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 7.7.82a
82. Use the definitions of the hyperbolic functions to find each of the following limits.
a. lim(x→∞) tanh x
Verified step by step guidance1
Recall the definition of the hyperbolic tangent function: \(\tanh x = \frac{\sinh x}{\cosh x}\).
Express \(\sinh x\) and \(\cosh x\) in terms of exponential functions: \(\sinh x = \frac{e^{x} - e^{-x}}{2}\) and \(\cosh x = \frac{e^{x} + e^{-x}}{2}\).
Substitute these into the expression for \(\tanh x\): \(\tanh x = \frac{\frac{e^{x} - e^{-x}}{2}}{\frac{e^{x} + e^{-x}}{2}} = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\).
Analyze the behavior of the numerator and denominator as \(x \to \infty\): since \(e^{x}\) grows very large and \(e^{-x}\) approaches zero, simplify the expression accordingly.
Use this simplification to find the limit \(\lim_{x \to \infty} \tanh x\) by considering dominant terms in numerator and denominator.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of Hyperbolic Functions
Hyperbolic functions such as sinh(x), cosh(x), and tanh(x) are defined using exponential functions: sinh(x) = (e^x - e^(-x))/2, cosh(x) = (e^x + e^(-x))/2, and tanh(x) = sinh(x)/cosh(x). Understanding these definitions is essential to analyze their behavior and limits.
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Definition of the Definite Integral
Limits Involving Exponential Functions
Evaluating limits as x approaches infinity often involves understanding the growth rates of exponential functions. Since e^x grows without bound and e^(-x) approaches zero as x → ∞, these behaviors help simplify expressions involving hyperbolic functions.
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Limit of tanh(x) as x Approaches Infinity
Using the definitions, tanh(x) = (e^x - e^(-x)) / (e^x + e^(-x)). As x → ∞, e^x dominates e^(-x), so tanh(x) approaches (∞ - 0)/(∞ + 0) = 1. Recognizing this helps find the limit without complex algebra.
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Related Practice
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?
a. x² + √x
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)
a. y = e^(-x)
Textbook Question
89. Use limits to find horizontal asymptotes for each function.
a. y = x tan(1/x)
Textbook Question
[Technology Exercise] In Exercises 139–141, find the domain and range of each composite function. Then graph the compositions on separate screens. Do the graphs make sense in each case? Give reasons for your answers. Comment on any differences you see.
141. a. y=arccos(cos x)
Textbook Question
154. The linearization of log₃x
a. Find the linearization of
f(x) = log₃xatx = 3.
Then round its coefficients to two decimal places.