82. Use the definitions of the hyperbolic functions to find each of the following limits.
a. lim(x→∞) tanh x

Verified step by step guidance

Recall the definition of the hyperbolic tangent function: \(\tanh x = \frac{\sinh x}{\cosh x}\).

Express \(\sinh x\) and \(\cosh x\) in terms of exponential functions: \(\sinh x = \frac{e^{x} - e^{-x}}{2}\) and \(\cosh x = \frac{e^{x} + e^{-x}}{2}\).

Substitute these into the expression for \(\tanh x\): \(\tanh x = \frac{\frac{e^{x} - e^{-x}}{2}}{\frac{e^{x} + e^{-x}}{2}} = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\).

Analyze the behavior of the numerator and denominator as \(x \to \infty\): since \(e^{x}\) grows very large and \(e^{-x}\) approaches zero, simplify the expression accordingly.

Use this simplification to find the limit \(\lim_{x \to \infty} \tanh x\) by considering dominant terms in numerator and denominator.

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Key Concepts

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

Definition of Hyperbolic Functions

Hyperbolic functions such as sinh(x), cosh(x), and tanh(x) are defined using exponential functions: sinh(x) = (e^x - e^(-x))/2, cosh(x) = (e^x + e^(-x))/2, and tanh(x) = sinh(x)/cosh(x). Understanding these definitions is essential to analyze their behavior and limits.

Limits Involving Exponential Functions

Evaluating limits as x approaches infinity often involves understanding the growth rates of exponential functions. Since e^x grows without bound and e^(-x) approaches zero as x → ∞, these behaviors help simplify expressions involving hyperbolic functions.

Limit of tanh(x) as x Approaches Infinity

Using the definitions, tanh(x) = (e^x - e^(-x)) / (e^x + e^(-x)). As x → ∞, e^x dominates e^(-x), so tanh(x) approaches (∞ - 0)/(∞ + 0) = 1. Recognizing this helps find the limit without complex algebra.

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