Problem 50 - Essential Calculus Early Transcendentals

Question

Evaluate \(\lim _{x \rightarrow \infty} \frac{\sinh x}{e^{x}}\)

Step-by-Step Solution

Answer

\( \lim_{x \to \infty} \frac{\sinh x}{e^x} = \frac{1}{2} \)

Step 1: Understand Hyperbolic Function

Recall that the hyperbolic sine function is defined as \( \sinh x = \frac{e^x - e^{-x}}{2} \). This will be useful for simplifying the limit expression.

Step 2: Substitute the Hyperbolic Function

Substitute \( \sinh x \) in the limit expression with its definition. This gives us \( \lim_{x \to \infty} \frac{\sinh x}{e^x} = \lim_{x \to \infty} \frac{\frac{e^x - e^{-x}}{2}}{e^x} \).

Step 3: Simplify the Fraction

Simplify the expression \( \frac{\frac{e^x - e^{-x}}{2}}{e^x} \) to \( \frac{e^x - e^{-x}}{2e^x} \). This further simplifies to \( \frac{e^x}{2e^x} - \frac{e^{-x}}{2e^x} \).

Step 4: Further Simplification

Divide through the fraction: \( \frac{1}{2} - \frac{e^{-2x}}{2} \). Notice \( e^{-2x} \rightarrow 0 \) as \( x \rightarrow \infty \).

Step 5: Evaluate the Limit

Evaluate the limit expression \( \lim_{x \to \infty} \left( \frac{1}{2} - \frac{e^{-2x}}{2} \right) \). Since \( \frac{e^{-2x}}{2} \to 0 \) as \( x \to \infty \), the limit simplifies to \( \frac{1}{2} \).

Key Concepts

Hyperbolic Functions

Hyperbolic functions are analogs of trigonometric functions but are based on hyperbolas instead of circles. The hyperbolic sine function, symbolized as \( \sinh x \), is particularly significant in calculus and is defined as \( \sinh x = \frac{e^x - e^{-x}}{2} \).
This definition involves exponential functions, which makes hyperbolic functions very versatile in various mathematical computations.
A unique aspect of hyperbolic functions like \( \sinh x \) is their ability to describe systems where growth is exponential. They can model real-world phenomena, such as the hanging chain or catenary, which forms the shape of hyperbolas.
Understanding hyperbolic sine is crucial in calculus, as it aids in the simplification of expression when finding limits or solving differential equations.
In the context of \( \lim _{x \rightarrow \infty} \frac{\sinh x}{e^{x}} \), we use the definition of \( \sinh x \) to simplify and evaluate limits at infinity.

Exponential Functions

Exponential functions are fundamental in various fields, notably in calculus and applied mathematics. They are functions whose growth rate is proportional to their value, described typically as \( e^x \).
Such functions are crucial in modeling continuous growth or decay phenomena, like population growth or radioactive decay.
The base of natural log, \( e \), approximately 2.71828, is used extensively because it simplifies the differentiation and integration processes in calculus.
When tackling the original exercise \( \lim _{x \rightarrow \infty} \frac{\sinh x}{e^{x}} \), the exponential nature of \( e^x \) plays a significant role.
As \( x \to \infty \), the \( e^x \) dominates any term that decreases exponentially, such as \( e^{-x} \).
This dominance makes exponential functions powerful in determining the behavior of functions as they approach infinity.

Calculus Concepts

Calculus is a branch of mathematics that focuses on change and motion. Two primary areas in calculus are differentiation and integration, which analyze the rate of change and the accumulation of quantities, respectively.
Limits are a fundamental concept in calculus, providing a way to understand how a function behaves as it approaches a particular point or infinity.
When solving limits involving hyperbolic and exponential functions, such as \( \lim _{x \rightarrow \infty} \frac{\sinh x}{e^{x}} \), recognizing the behavior of individual components is vital.

• First, substitute the hyperbolic function with its definition.
• Simplify the expression to see dominant terms as \( x \) approaches infinity.
• Identify parts of the expression that approach zero, leveraging the properties of exponential functions.

The step-by-step evaluation involves simplifying complex expressions and effectively applying the concept of limits.
Mastering such techniques is instrumental in solving problems where the behavior of equations is analyzed as inputs grow indefinitely.