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

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

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

Problema 7.R.32

Capitolo 7, Problema 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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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.

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Integrals of Natural Exponential Functions (e^x)

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.

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Integrals Involving Natural Logs: Substitution

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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Evaluate Logarithms

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