82. Use the definitions of the hyperbolic functions to find each of the following limits.
i. lim(x→-∞) csch x

Verified step by step guidance

1. Recall the definition of the hyperbolic cosecant function: \(\text{csch}\,x = \frac{1}{\sinh x}\).
2. Recall the definition of the hyperbolic sine function: \(\sinh x = \frac{e^{x} - e^{-x}}{2}\).
3. Analyze the behavior of \(\sinh x\) as \(x \to -\infty\). Since \(e^{x} \to 0\) and \(e^{-x} \to \infty\) as \(x \to -\infty\), determine the dominant term in \(\sinh x\).
4. Use the dominant term to approximate \(\sinh x\) for very large negative \(x\), then find the corresponding behavior of \(\text{csch}\,x = \frac{1}{\sinh x}\).
5. Conclude the limit \(\lim_{x \to -\infty} \text{csch}\,x\) based on the sign and magnitude of the approximation.

Key Concepts

Definition of Hyperbolic Cosecant (csch x)

The hyperbolic cosecant function, csch x, is defined as 1 divided by the hyperbolic sine of x, i.e., csch x = 1/sinh x. Understanding this definition is essential to rewrite the limit expression in terms of exponential functions for easier evaluation.

Behavior of Exponential Functions as x Approaches Negative Infinity

As x approaches negative infinity, the exponential function e^x approaches zero, while e^{-x} grows without bound. Recognizing this behavior helps simplify expressions involving hyperbolic functions, which are combinations of exponentials, to determine their limits.

Limit Evaluation Using Exponential Definitions

By expressing hyperbolic functions in terms of exponentials, limits can be evaluated by analyzing dominant terms as x approaches infinity or negative infinity. This method allows for straightforward calculation of limits by focusing on the leading exponential behavior.