The hyperbolic cosine function, denoted $\cosh \left(x\right)$, is used to model the shape of a hanging cable (a telephone wire, for example). It is defined as $\cosh \left(x\right)=\frac{e^x+e^{-x}}{2}$.


a. Determine its end behavior by analyzing $\underset{x\to \infty }{\lim }\cosh \left(x\right)$ and $\underset{x\to -\infty }{\lim }\cosh \left(x\right)$.

Verified step by step guidance

1. Step 1: Recall the definition of the hyperbolic cosine function: \( \cosh(x) = \frac{e^x + e^{-x}}{2} \).
2. Step 2: To find the end behavior as \( x \to \infty \), consider the terms \( e^x \) and \( e^{-x} \). As \( x \to \infty \), \( e^x \) grows very large while \( e^{-x} \) approaches zero.
3. Step 3: Therefore, as \( x \to \infty \), \( \cosh(x) \approx \frac{e^x}{2} \), which implies that \( \cosh(x) \to \infty \).
4. Step 4: Now, consider the behavior as \( x \to -\infty \). In this case, \( e^x \) approaches zero and \( e^{-x} \) grows very large.
5. Step 5: Thus, as \( x \to -\infty \), \( \cosh(x) \approx \frac{e^{-x}}{2} \), which also implies that \( \cosh(x) \to \infty \).

Key Concepts

Hyperbolic Functions

Hyperbolic functions, such as the hyperbolic cosine (cosh), are analogs of trigonometric functions but are based on hyperbolas instead of circles. The hyperbolic cosine function is defined as cosh(x) = (e^x + e^(-x))/2, where e is the base of the natural logarithm. These functions are useful in various applications, including modeling shapes and phenomena in physics and engineering.

Limits

Limits are a fundamental concept in calculus that describe the behavior of a function as its input approaches a certain value. In this context, we analyze the limits of the hyperbolic cosine function as x approaches positive and negative infinity. Understanding limits helps determine the end behavior of functions, which is crucial for graphing and analyzing their properties.

End Behavior of Functions

The end behavior of a function refers to how the function behaves as the input values approach positive or negative infinity. For the hyperbolic cosine function, analyzing its limits at infinity reveals that it grows without bound as x approaches positive infinity and approaches a constant value as x approaches negative infinity. This information is essential for understanding the overall shape and characteristics of the function's graph.