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?
e. x^3 - x^2
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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 7.7.82f
82. Use the definitions of the hyperbolic functions to find each of the following limits.
f. lim(x→∞) coth x
Guida verificata passo dopo passo1
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: \(\lim_{x \to \infty} \coth x = \lim_{x \to \infty} \frac{1 + e^{-2x}}{1 - e^{-2x}}\).
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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.
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Percorso guidato
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.
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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.
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05:21Finding Limits by Direct Substitution
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