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

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

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

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

As \(x \to \infty\), analyze the behavior of \(e^{x}\) and \(e^{-x}\): \(e^{x} \to \infty\) and \(e^{-x} \to 0\).

Simplify the expression by dividing numerator and denominator by \(e^{x}\) to find the limit: \(\lim_{x \to \infty} \coth x = \lim_{x \to \infty} \frac{1 + e^{-2x}}{1 - e^{-2x}}\).

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Definition of Hyperbolic Functions

Hyperbolic functions such as sinh x, cosh x, and coth x are defined using exponential functions. Specifically, coth x is defined as cosh x divided by sinh x, where cosh x = (e^x + e^{-x})/2 and sinh x = (e^x - e^{-x})/2. Understanding these definitions is essential to analyze limits involving hyperbolic functions.

Limit of Exponential Functions as x Approaches Infinity

As x approaches infinity, e^x grows without bound while e^{-x} approaches zero. This behavior simplifies expressions involving exponentials, allowing us to approximate hyperbolic functions for large x values. Recognizing these limits helps in evaluating the limit of coth x as x tends to infinity.

Evaluating Limits Using Algebraic Simplification

To find limits involving hyperbolic functions, it is often necessary to rewrite the expression using their exponential definitions and simplify. This process includes factoring, canceling terms, and applying known limits of exponential functions. Mastery of these algebraic techniques is crucial for correctly determining the limit of coth x as x approaches infinity.

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