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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 7.3.91
Chapter 7, Problem 7.3.91

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


lim x → 0⁺ (tanh x)ˣ

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1
Recognize that the limit is of the form \(\lim_{x \to 0^+} (\tanh x)^x\). Since \(\tanh x\) approaches 0 as \(x\) approaches 0, and the exponent \(x\) also approaches 0, this is an indeterminate form of type \$0^0$.
Rewrite the expression using the exponential and logarithm functions to handle the indeterminate form: \(\lim_{x \to 0^+} (\tanh x)^x = \lim_{x \to 0^+} e^{x \ln(\tanh x)}\).
Focus on evaluating the exponent limit: \(\lim_{x \to 0^+} x \ln(\tanh x)\). This is a product of \(x\) and \(\ln(\tanh x)\), which tends to \(0 \cdot (-\infty)\), another indeterminate form.
Rewrite the product as a quotient to apply l'Hôpital's Rule: \(\lim_{x \to 0^+} \frac{\ln(\tanh x)}{1/x}\). Now, as \(x \to 0^+\), the numerator tends to \(-\infty\) and the denominator tends to \(+\infty\), so the limit is of the form \(\frac{-\infty}{\infty}\).
Apply l'Hôpital's Rule by differentiating numerator and denominator with respect to \(x\): differentiate \(\ln(\tanh x)\) and \(1/x\), then take the limit of their ratio as \(x \to 0^+\). This will give the limit of the exponent, which you can then use to find the original limit.

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Key Concepts

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

Limits and Continuity

Limits describe the behavior of a function as the input approaches a particular value. Understanding how to evaluate limits, especially at points where direct substitution leads to indeterminate forms, is essential for analyzing function behavior near those points.
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Indeterminate Forms

Indeterminate forms like 0^0, ∞/∞, or 0/0 occur when direct substitution in a limit does not yield a clear value. Recognizing these forms is crucial because they signal the need for special techniques, such as algebraic manipulation or applying l’Hôpital’s Rule, to evaluate the limit.
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l’Hôpital’s Rule

l’Hôpital’s Rule helps evaluate limits that result in indeterminate forms 0/0 or ∞/∞ by differentiating the numerator and denominator separately. Applying this rule correctly simplifies complex limits, making it easier to find the limit’s value when direct substitution fails.
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