Textbook Question
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ sin(2x) cos(4x) 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 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 7.7.82i
82. Use the definitions of the hyperbolic functions to find each of the following limits.
i. lim(x→-∞) csch x
Verified step by step guidance1
Recall the definition of the hyperbolic cosecant function: \(\text{csch}\,x = \frac{1}{\sinh x}\).
Recall the definition of the hyperbolic sine function: \(\sinh x = \frac{e^{x} - e^{-x}}{2}\).
Analyze the behavior of \(\sinh x\) as \(x \to -\infty\). Since \(e^{x} \to 0\) and \(e^{-x} \to \infty\) as \(x \to -\infty\), determine the dominant term in \(\sinh x\).
Use the dominant term to approximate \(\sinh x\) for very large negative \(x\), then find the corresponding behavior of \(\text{csch}\,x = \frac{1}{\sinh x}\).
Conclude the limit \(\lim_{x \to -\infty} \text{csch}\,x\) based on the sign and magnitude of the approximation.
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 Cosecant (csch x)
The hyperbolic cosecant function, csch x, is defined as 1 divided by the hyperbolic sine of x, i.e., csch x = 1/sinh x. Understanding this definition is essential to rewrite the limit expression in terms of exponential functions for easier evaluation.
Recommended video:
05:43
Definition of the Definite Integral
Behavior of Exponential Functions as x Approaches Negative Infinity
As x approaches negative infinity, the exponential function e^x approaches zero, while e^{-x} grows without bound. Recognizing this behavior helps simplify expressions involving hyperbolic functions, which are combinations of exponentials, to determine their limits.
Recommended video:
5:46Graphs of Exponential Functions
Limit Evaluation Using Exponential Definitions
By expressing hyperbolic functions in terms of exponentials, limits can be evaluated by analyzing dominant terms as x approaches infinity or negative infinity. This method allows for straightforward calculation of limits by focusing on the leading exponential behavior.
Recommended video:
5:47Solving Exponential Equations Using Logs
Related Practice
Textbook Question
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (tan²x + sec²x) dx
Textbook Question
82. Use the definitions of the hyperbolic functions to find each of the following limits.
h. lim(x→0^-) coth x
1
views
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
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ sin(2x) cos(3x) dx
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