82. Use the definitions of the hyperbolic functions to find each of the following limits.
h. lim(x→0^-) coth x

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Recall the definition of the hyperbolic cotangent function: \(\coth x = \frac{\cosh x}{\sinh x}\), where \(\cosh x = \frac{e^{x} + e^{-x}}{2}\) and \(\sinh x = \frac{e^{x} - e^{-x}}{2}\).

Express \(\coth x\) explicitly in terms of exponentials: \(\coth x = \frac{\frac{e^{x} + e^{-x}}{2}}{\frac{e^{x} - e^{-x}}{2}} = \frac{e^{x} + e^{-x}}{e^{x} - e^{-x}}\).

Analyze the behavior of the numerator and denominator as \(x\) approaches \(0\) from the left (i.e., \(x \to 0^-\)). Consider the series expansions or the values of \(e^{x}\) and \(e^{-x}\) near zero.

Since both numerator and denominator approach zero, consider simplifying the expression or using limits properties such as L'Hôpital's Rule if necessary.

Apply the limit \(\lim_{x \to 0^-} \coth x\) by substituting the expressions and evaluating the limit carefully, keeping in mind the sign of \(x\) approaching zero from the negative side.

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Definition of Hyperbolic Cotangent (coth x)

The hyperbolic cotangent function, coth x, is defined as the ratio of the hyperbolic cosine to the hyperbolic sine: coth x = cosh x / sinh x. Understanding this definition is essential to analyze its behavior near specific points, such as x approaching zero.

Behavior of Hyperbolic Sine and Cosine Near Zero

Near x = 0, sinh x behaves like x (since sinh x ≈ x for small x), and cosh x approaches 1. This approximation helps simplify the limit expressions involving hyperbolic functions and is crucial for evaluating limits as x approaches zero.

Limit from the Left-Hand Side (x → 0⁻)

When evaluating limits as x approaches zero from the left (x → 0⁻), it is important to consider the sign and behavior of the function values just less than zero. For coth x, since sinh x changes sign around zero, the left-hand limit may differ from the right-hand limit.

82. Use the definitions of the hyperbolic functions to find each of the following limits.h. lim(x→0^-) coth x