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

Briggs 3rd Edition

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

Problem 7.R.32

Chapter 7, Problem 7.R.32

Limit Evaluate lim x → ∞ (tanh x)ˣ.

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Recall the definition of the hyperbolic tangent function: \(\tanh x = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\).

Analyze the behavior of \(\tanh x\) as \(x \to \infty\). Since \(e^{x}\) grows much faster than \(e^{-x}\), \(\tanh x\) approaches 1 from below.

Express the limit in the form \(\lim_{x \to \infty} (\tanh x)^{x} = \lim_{x \to \infty} \left(1 - \epsilon_x\right)^{x}\), where \(\epsilon_x\) is a small positive number approaching 0 as \(x \to \infty\).

Use the approximation \(\ln(1 - \epsilon_x) \approx -\epsilon_x\) for small \(\epsilon_x\) to rewrite the limit as \(\lim_{x \to \infty} e^{x \ln(\tanh x)} \approx \lim_{x \to \infty} e^{-x \epsilon_x}\).

Determine the rate at which \(\epsilon_x\) approaches 0 to evaluate the limit of the exponent \(-x \epsilon_x\) and thus find the behavior of the original limit.

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Key Concepts

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

Behavior of the Hyperbolic Tangent Function (tanh x) as x → ∞

The hyperbolic tangent function, tanh x, approaches 1 as x approaches infinity. Understanding this limit is crucial because it determines the base of the expression (tanh x)^x for large x values.

Limits Involving Indeterminate Forms

Expressions like (tanh x)^x can lead to indeterminate forms such as 1^∞. Recognizing and resolving these forms often requires rewriting the expression using logarithms or other techniques to evaluate the limit accurately.

Use of Logarithms to Evaluate Limits of Exponential Forms

Taking the natural logarithm of an expression like (tanh x)^x transforms it into x * ln(tanh x), simplifying the limit evaluation. After finding the limit of the logarithm, exponentiate the result to obtain the original limit.

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Textbook Question

Linear approximation Find the linear approximation to ƒ(x) = cosh x at a = ln 3 and then use it to approximate the value of cosh 1.

Textbook Question

Log-normal probability distribution A commonly used distribution in probability and statistics is the log-normal distribution. (If the logarithm of a variable has a normal distribution, then the variable itself has a log-normal distribution.) The distribution function is

f(x) = 1/xσ√(2π) e⁻ˡⁿ^² ˣ / ²σ^², for x ≥ 0

where ln x has zero mean and standard deviation σ > 0.

e. For what value of σ > 0 in part (d) does ƒ(x*) have a minimum?

Textbook Question

Derivatives of hyperbolic functions Compute the following derivatives.

b. d/dx (x sech x)

Textbook Question

10–19. Derivatives Find the derivatives of the following functions.

f(t) = cosh t sinh t

Textbook Question

10–19. Derivatives Find the derivatives of the following functions.

f(x) = tanh⁻¹(cos x)

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. ln xy = (ln x)(ln y)