Problem 113 - Calculus and its applications

Question

Graph $$ f(x)=\left(1+\frac{1}{x}\right)^{x} $$ Use the TABLE feature and very large values of \(x\) to confirm that \(e\) is approached as a limit.

Step-by-Step Solution

Answer

As \( x \to \infty \), \( f(x) \to e \).

Step 1: Understand the Function

The function is given as \( f(x) = \left(1 + \frac{1}{x}\right)^{x} \). This is a classic expression that approaches the mathematical constant \( e \) as \( x \) becomes very large.

Step 2: Set Up the TABLE Feature

Open a graphing calculator or software that supports a table feature. Input the function \( f(x) = \left(1 + \frac{1}{x}\right)^{x} \) to generate its values for different inputs.

Step 3: Choose Values for \( x \)

Select very large values for \( x \) such as 10, 100, 1000, 10000, etc. in the table. The goal is to observe how the function behaves as \( x \) increases.

Step 4: Evaluate the Function

For each chosen value of \( x \), compute \( f(x) \) using the table feature. Record these values to see the trend.

Step 5: Analyze the Results

Look at the calculated values of \( f(x) \) for increasing \( x \). Notice how the results get closer to the value of \( e \), which is approximately 2.718.

Step 6: Conclusion

The table shows that as \( x \) becomes very large, \( f(x) \) approaches \( e \). This confirms that \( \lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^{x} = e \).

Key Concepts

Exponential Functions

An exponential function is a type of mathematical function, where one of the variables appears as the exponent. In our exercise, the function is \( f(x) = \left(1 + \frac{1}{x}\right)^{x} \), where the expression within the brackets \( 1 + \frac{1}{x} \) is raised to the power of \( x \). This is an example of how exponential functions can model growth or decay, depending on the scenario. Exponential functions are significant because:

• They describe processes that grow or shrink at a constant rate.
• They are used in various fields such as biology, finance, and physics.

In the case of our function, as \( x \) becomes very large, the term \( \frac{1}{x} \) becomes very small, thereby making \( 1 + \frac{1}{x} \) approach 1. However, since \( x \) is also in the base, it results in a value that converges to the mathematical constant \( e \). This convergence is an essential characteristic of this type of exponential function.

Graphing Calculator

Graphing calculators are powerful tools used to visualize mathematical functions. They allow us to input complex expressions and observe their behavior over different variables. In this exercise, a graphing calculator is used to analyze how the function \( f(x) = \left(1 + \frac{1}{x}\right)^{x} \) behaves as \( x \) increases. Here's how a graphing calculator can help:

• It can create a table of values for different inputs of \( x \).
• It allows for quick computation and visualization of functions.
• It helps in spotting trends, making it easier to understand asymptotic behavior.

Using the calculator's table feature, you can input large values of \( x \) and observe that the corresponding function values approach \( e \). This type of analysis is very effective for learning and confirming mathematical concepts.

Mathematical Constant e

The mathematical constant \( e \) is one of the most important numbers in mathematics, approximately equal to 2.718. It is the base of natural logarithms and appears frequently in calculus and mathematical analysis.Some key points about \( e \):

• It is an irrational number, meaning it cannot be expressed precisely as a fraction.
• It is the limit of \( \left(1 + \frac{1}{x}\right)^{x} \) as \( x \) approaches infinity.
• It is often used in calculations involving growth and decay processes, such as compound interest and population growth.

In our exercise, as we input larger and larger values of \( x \) into \( \left(1 + \frac{1}{x}\right)^{x} \), we observe the behavior of \( e \). This exercise illustrates how \( e \) emerges naturally from certain types of exponential growth, making it a fundamental component in advanced mathematical concepts.