Question

In Exercises \(63-68,\) find the limit. \(\lim _{x \rightarrow \infty} \sinh x\)

Short answer

The limit of the hyperbolic sine function as \(x\) approaches infinity is infinity (\(\infty\)).

Step-by-step solution

Understand the hyperbolic sine function

The hyperbolic sine function, denoted as \(\sinh x\), can be defined in terms of exponential functions as \(\sinh x = \frac{e^x - e^{-x}}{2}\). So, the limit problem can be rewritten as: \(\lim_{x \rightarrow \infty} \frac{e^x - e^{-x}}{2}\).

Calculate the limit

As \(x\) approaches infinity, \(e^x\) also approaches infinity, while \(e^{-x}\) approaches zero. This due to the laws of exponents (anything raised to the power \(-x\) is equivalent to 1/(the same thing raised to the power \(x\))). So, the limit can be rewritten as \(\lim_{x \rightarrow \infty} \frac{e^x - 0}{2}\), which simplifies to \(\lim_{x \rightarrow \infty} \frac{e^x}{2}\). This limit approaches infinity as \(x\) approaches infinity.

Conclusion

Therefore, we can conclude that \(\lim_{x \rightarrow \infty} \sinh x = \infty\).

Key concepts

Hyperbolic Functions

Hyperbolic functions are analogous to the trigonometric functions but for a hyperbola, as trigonometric functions are to the circle. They are defined through exponential functions and have numerous properties that make them useful in various branches of mathematics, including calculus.

For instance, the hyperbolic sine function, denoted as \( \sinh x \), is given by \( \sinh x = \frac{e^x - e^{-x}}{2} \). This function graph mirrors the shape of a hanging cable, such as a power line, which makes it relevant for real-world applications in engineering and physics. When finding limits of hyperbolic functions, it's essential to understand their relationship with exponential growth and decay.

Exponential Functions

Exponential functions are fundamental in mathematics, representing continuous growth or decay. They are written in the form \( f(x) = a^x \), where \( a \) is a positive constant base, and \( x \) is the exponent. Such functions are crucial because they model phenomena in fields like finance, biology, and physics.

The limit behavior of exponential functions is especially interesting. As the exponent grows large, the function can grow without bound if the base is greater than 1. Conversely, if we have a negative exponent, the function gets closer and closer to zero, but never quite reaches it. This ties directly into understanding the limit of \( \sinh x \) as \( x \) approaches infinity.

Laws of Exponents

The laws of exponents are a set of rules that describe how to handle mathematical operations involving exponents. They're vital for simplifying expressions and understanding the behavior of exponential functions. Several of these rules directly impact the evaluation of limits in calculus:

• Product of Powers: \( a^m \cdot a^n = a^{m+n} \)
• Quotient of Powers: \( a^m / a^n = a^{m-n} \)
• Power of a Power: \( (a^m)^n = a^{mn} \)
• Power of a Product: \( (ab)^m = a^m b^m \)
• Zero Exponent: \( a^0 = 1 \)
• Negative Exponent: \( a^{-n} = 1/a^n \)

These rules enable us to manipulate and simplify exponential expressions, particularly when dealing with limits of functions as variables approach infinity.

Limits at Infinity

Analyzing the behavior of functions as variables approach infinity, called 'limits at infinity', is a cornerstone of calculus. It helps us understand how functions behave towards the ends of the number line or as time goes to infinity. When we say \( \lim_{x \rightarrow \infty} f(x) \) equals some value \( L \), we mean that as \( x \) gets larger and larger, the function \( f(x) \) gets closer and closer to the value \( L \).

In the case of the hyperbolic sine function, as \( x \) trends towards infinity, \( e^x \) overwhelms the \( e^{-x} \) term because the latter approaches zero. Hence, \( \lim_{x \rightarrow \infty} \sinh x \) grows without bound, which we express as \( \infty \). This kind of analysis is crucial for predicting the long-term behavior of dynamic systems, which is why studying limits is so valuable.