Skip to main content
Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.7.82i
Capitolo 7, Problema 7.7.82i

82. Use the definitions of the hyperbolic functions to find each of the following limits.
i. lim(x→-∞) csch x

Guida verificata passo dopo passo
1
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.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.
Video consigliato:
Percorso guidato
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.
Video consigliato:
5:46
Graphs 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.
Video consigliato:
5:47
Solving Exponential Equations Using Logs
Pratica correlata
Domanda del libro di testo

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ sin(2x) cos(4x) dx

21
views
Domanda del libro di testo

9. True, or false? As x→∞,

g. ln(x) = o(ln(2x))

32
views
Domanda del libro di testo

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

21
views
Domanda del libro di testo

82. Use the definitions of the hyperbolic functions to find each of the following limits.

h. lim(x→0^-) coth x

33
views
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

27
views
Domanda del libro di testo

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ sin(2x) cos(3x) dx

25
views