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

검증된 단계별 안내

Recall the definition of the hyperbolic sine function: \(\sinh x = \frac{e^{x} - e^{-x}}{2}\).

Rewrite the limit using this definition: \(\lim_{x \to \infty} \sinh x = \lim_{x \to \infty} \frac{e^{x} - e^{-x}}{2}\).

Analyze the behavior of each term inside the limit as \(x\) approaches infinity: \(e^{x}\) grows without bound, while \(e^{-x}\) approaches zero.

Since \(e^{x}\) dominates \(e^{-x}\) for large \(x\), the expression behaves like \(\frac{e^{x}}{2}\) as \(x \to \infty\).

Conclude that the limit depends on the growth of \(e^{x}\), which increases without bound, so the limit tends toward infinity.

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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.

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