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 \( \cosh(x) \to \infty \).