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

1. Recall the definition of the hyperbolic tangent function: \(\tanh x = \frac{\sinh x}{\cosh x}\).
2. Express \(\sinh x\) and \(\cosh x\) in terms of exponential functions: \(\sinh x = \frac{e^{x} - e^{-x}}{2}\) and \(\cosh x = \frac{e^{x} + e^{-x}}{2}\).
3. Substitute these into the expression for \(\tanh x\): \(\tanh x = \frac{\frac{e^{x} - e^{-x}}{2}}{\frac{e^{x} + e^{-x}}{2}} = \frac{e^{x} - e^{-x}}{e^{x} + e^{-x}}\).
4. 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.
5. Use this simplification to find the limit \(\lim_{x \to \infty} \tanh x\) by considering dominant terms in numerator and denominator.

Key Concepts

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.