Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo

Briggs 3rd Edition

Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions

Problema 7.3.88

Capitolo 7, Problema 7.3.88

88–91. Limits Use l’Hôpital’s Rule to evaluate the following limits.

lim x → ∞ (1 − coth x) / (1 − tanh x)

Guida verificata passo dopo passo

Identify the limit expression: \(\lim_{x \to \infty} \frac{1 - \coth x}{1 - \tanh x}\).

Recall the definitions of hyperbolic functions: \(\coth x = \frac{\cosh x}{\sinh x}\) and \(\tanh x = \frac{\sinh x}{\cosh x}\).

Evaluate the behavior of \(\coth x\) and \(\tanh x\) as \(x \to \infty\) to check if the limit is an indeterminate form suitable for l'Hôpital's Rule.

If the limit is of the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), apply l'Hôpital's Rule by differentiating numerator and denominator separately with respect to \(x\):

Compute \(\frac{d}{dx} (1 - \coth x)\) and \(\frac{d}{dx} (1 - \tanh x)\), then form the new limit \(\lim_{x \to \infty} \frac{\frac{d}{dx} (1 - \coth x)}{\frac{d}{dx} (1 - \tanh x)}\) and analyze this limit.

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Hyperbolic Functions (coth and tanh)

Hyperbolic functions like coth(x) and tanh(x) are analogs of trigonometric functions but based on hyperbolas. As x approaches infinity, tanh(x) approaches 1, while coth(x) also approaches 1, but understanding their behavior is crucial for evaluating limits involving these functions.

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Percorso guidato

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Asymptotes of Hyperbolas

Limits at Infinity

Evaluating limits as x approaches infinity involves analyzing the behavior of functions for very large values of x. Recognizing how functions like coth(x) and tanh(x) behave at infinity helps determine the form of the limit and whether it is indeterminate.

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Cases Where Limits Do Not Exist

l’Hôpital’s Rule

l’Hôpital’s Rule is used to evaluate limits that result in indeterminate forms like 0/0 or ∞/∞ by differentiating the numerator and denominator separately. Applying this rule simplifies complex limits, especially when direct substitution leads to indeterminate expressions.

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Power Rules

Pratica correlata

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37–56. Integrals Evaluate each integral.

∫ dx/x√(16 + x²)

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29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫ (x²) / (4x³ + 7) dx

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Surface area of a catenoid When the catenary y = a cosh x/a is revolved about the x-axis, it sweeps out a surface of revolution called a catenoid. Find the area of the surface generated when y = cosh x on [–ln 2, ln 2] is rotated about the x-axis.

120

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63–66. Calculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with l’Hôpital’s Rule.

limₕ→₀ (1 + 3h)^{2/h}

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22–36. Derivatives Find the derivatives of the following functions.

f(x) = x sinh⁻¹ x − √(x² + 1)

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37–56. Integrals Evaluate each integral.

∫ sech² w tanh w dw

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