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Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ sin(2x) cos(4x) dx
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 7, Problema 7.7.82h
82. Use the definitions of the hyperbolic functions to find each of the following limits.
h. lim(x→0^-) coth x
Guida verificata passo dopo passo1
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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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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Percorso guidato
05:43
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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03:53Derivatives of Sine & Cosine
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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05:50One-Sided Limits
Pratica correlata
Domanda del libro di testo
9. True, or false? As x→∞,
g. ln(x) = o(ln(2x))
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Domanda del libro di testo
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
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Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ sin(2x) cos(3x) dx
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Domanda del libro di testo
82. Use the definitions of the hyperbolic functions to find each of the following limits.
i. lim(x→-∞) csch x
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Domanda del libro di testo
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
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