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
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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 7.7.82h
82. Use the definitions of the hyperbolic functions to find each of the following limits.
h. lim(x→0^-) coth x
Verified step by step guidance1
Recall the definition of the hyperbolic cotangent function: \(\coth x = \frac{\cosh x}{\sinh x}\), where \(\cosh x = \frac{e^{x} + e^{-x}}{2}\) and \(\sinh x = \frac{e^{x} - e^{-x}}{2}\).
Express \(\coth x\) explicitly in terms of exponentials: \(\coth x = \frac{\frac{e^{x} + e^{-x}}{2}}{\frac{e^{x} - e^{-x}}{2}} = \frac{e^{x} + e^{-x}}{e^{x} - e^{-x}}\).
Analyze the behavior of the numerator and denominator as \(x\) approaches \(0\) from the left (i.e., \(x \to 0^-\)). Consider the series expansions or the values of \(e^{x}\) and \(e^{-x}\) near zero.
Since both numerator and denominator approach zero, consider simplifying the expression or using limits properties such as L'Hôpital's Rule if necessary.
Apply the limit \(\lim_{x \to 0^-} \coth x\) by substituting the expressions and evaluating the limit carefully, keeping in mind the sign of \(x\) approaching zero from the negative side.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definition of Hyperbolic Cotangent (coth x)
The hyperbolic cotangent function, coth x, is defined as the ratio of the hyperbolic cosine to the hyperbolic sine: coth x = cosh x / sinh x. Understanding this definition is essential to analyze its behavior near specific points, such as x approaching zero.
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Definition of the Definite Integral
Behavior of Hyperbolic Sine and Cosine Near Zero
Near x = 0, sinh x behaves like x (since sinh x ≈ x for small x), and cosh x approaches 1. This approximation helps simplify the limit expressions involving hyperbolic functions and is crucial for evaluating limits as x approaches zero.
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Limit from the Left-Hand Side (x → 0⁻)
When evaluating limits as x approaches zero from the left (x → 0⁻), it is important to consider the sign and behavior of the function values just less than zero. For coth x, since sinh x changes sign around zero, the left-hand limit may differ from the right-hand limit.
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Related Practice
Textbook Question
9. True, or false? As x→∞,
g. ln(x) = o(ln(2x))
1
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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)
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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
82. Use the definitions of the hyperbolic functions to find each of the following limits.
i. lim(x→-∞) csch x
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