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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.88
Chapter 7, Problem 7.3.88

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


lim x → ∞ (1 − coth x) / (1 − tanh x)

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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Related Practice
Textbook Question

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


∫ (x²) / (4x³ + 7) dx

Textbook Question

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.

Textbook Question

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}

Textbook Question

Probability as an integral Two points P and Q are chosen randomly, one on each of two adjacent sides of a unit square (see figure). What is the probability that the area of the triangle formed by the sides of the square and the line segment PQ is less than one-fourth the area of the square? Begin by showing that x and y must satisfy xy < 1/2 in order for the area condition to be met. Then argue that the required probability is: 1/2 + ∫[1/2 to 1] (dx / 2x) and evaluate the integral.

Textbook Question

Rule of 70 Bankers use the Rule of 70, which says that if an account increases at a fixed rate of p%/yr, its doubling time is approximately 70/p. Use linear approximation to explain why and when this is true.

Textbook Question

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


f(x) = x sinh⁻¹ x − √(x² + 1)