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
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 7.7.82f
82. Use the definitions of the hyperbolic functions to find each of the following limits.
f. lim(x→∞) coth x
Verified step by step guidance1
Recall the definition of the hyperbolic cotangent function: \(\coth x = \frac{\cosh x}{\sinh x}\).
Express \(\cosh x\) and \(\sinh x\) in terms of exponential functions: \(\cosh x = \frac{e^{x} + e^{-x}}{2}\) and \(\sinh x = \frac{e^{x} - e^{-x}}{2}\).
Substitute these into the expression for \(\coth x\): \(\coth x = \frac{\frac{e^{x} + e^{-x}}{2}}{\frac{e^{x} - e^{-x}}{2}} = \frac{e^{x} + e^{-x}}{e^{x} - e^{-x}}\).
As \(x \to \infty\), analyze the behavior of \(e^{x}\) and \(e^{-x}\): \(e^{x} \to \infty\) and \(e^{-x} \to 0\).
Simplify the expression by dividing numerator and denominator by \(e^{x}\) to find the limit: \(\lim_{x \to \infty} \coth x = \lim_{x \to \infty} \frac{1 + e^{-2x}}{1 - e^{-2x}}\).
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.
Definition of Hyperbolic Functions
Hyperbolic functions such as sinh x, cosh x, and coth x are defined using exponential functions. Specifically, coth x is defined as cosh x divided by sinh x, where cosh x = (e^x + e^{-x})/2 and sinh x = (e^x - e^{-x})/2. Understanding these definitions is essential to analyze limits involving hyperbolic functions.
Recommended video:
05:43
Definition of the Definite Integral
Limit of Exponential Functions as x Approaches Infinity
As x approaches infinity, e^x grows without bound while e^{-x} approaches zero. This behavior simplifies expressions involving exponentials, allowing us to approximate hyperbolic functions for large x values. Recognizing these limits helps in evaluating the limit of coth x as x tends to infinity.
Recommended video:
5:46Graphs of Exponential Functions
Evaluating Limits Using Algebraic Simplification
To find limits involving hyperbolic functions, it is often necessary to rewrite the expression using their exponential definitions and simplify. This process includes factoring, canceling terms, and applying known limits of exponential functions. Mastery of these algebraic techniques is crucial for correctly determining the limit of coth x as x approaches infinity.
Recommended video:
05:21Finding Limits by Direct Substitution
Related Practice
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
2. Express the following logarithms in terms of ln 5 and ln 7.
f. (ln35 + ln(1/7))/(ln25)
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
1. Express the following logarithms in terms of ln 2 and ln 3.
f. ln √(13.5)
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))