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

Verified step by step guidance

1. Recall the definition of the hyperbolic sine function: \(\sinh x = \frac{e^{x} - e^{-x}}{2}\).
2. Rewrite the limit using this definition: \(\lim_{x \to \infty} \sinh x = \lim_{x \to \infty} \frac{e^{x} - e^{-x}}{2}\).
3. Analyze the behavior of each term inside the limit as \(x\) approaches infinity: \(e^{x}\) grows without bound, while \(e^{-x}\) approaches zero.
4. Since \(e^{x}\) dominates \(e^{-x}\) for large \(x\), the expression behaves like \(\frac{e^{x}}{2}\) as \(x \to \infty\).
5. Conclude that the limit depends on the growth of \(e^{x}\), which increases without bound, so the limit tends toward infinity.

Key Concepts

Definition of Hyperbolic Sine Function

The hyperbolic sine function, sinh x, is defined as (e^x - e^(-x)) / 2. Understanding this definition is crucial because it expresses sinh x in terms of exponential functions, which simplifies the evaluation of limits involving sinh x.

Behavior of Exponential Functions at Infinity

As x approaches infinity, e^x grows without bound, while e^(-x) approaches zero. Recognizing this behavior helps in simplifying expressions like sinh x by focusing on dominant terms when evaluating limits at infinity.

Limit Evaluation Techniques

Evaluating limits often involves identifying dominant terms and applying limit laws. For sinh x as x approaches infinity, this means simplifying the expression using the exponential growth rates to determine the limit accurately.