Problem 56 - Algebra and Trigonometry

Question

Investigate the behavior of the function $$ f(x)=\left(1-\frac{1}{x}\right)^{x} $$ as \(x \rightarrow \infty\) by graphing \(f\) and the line \(y=1 / e\) on the same screen using the viewing rectangle \([0,20]\) by \([0,1].\)

Step-by-Step Solution

Answer

As \(x \to \infty\), \(f(x)\) approaches \(\frac{1}{e}\).

Step 1: Understand the Function

The function we are investigating is given by \( f(x) = \left( 1 - \frac{1}{x} \right)^x \). It resembles the exponential limit definition for \( e^{-1} \). Our task is to determine the behavior of this function as \( x \to \infty \).

Step 2: Graphing the Function

To understand the behavior visually, plot \( f(x) = \left( 1 - \frac{1}{x} \right)^x \) on a graph with \( x \) values ranging from 0 to 20 and \( y \) values ranging from 0 to 1. This helps visualize how \( f(x) \) changes as \( x \) increases.

Step 3: Add Reference Line \(y = 1/e\)

On the same graph, draw the horizontal line \( y = \frac{1}{e} \) to provide a reference of what \( f(x) \) is expected to approach, where \( e \) is the base of the natural logarithm (approximately 2.718). This line acts as a benchmark to compare the behavior of \( f(x) \).

Step 4: Analyze the Graph

Observe the plotted graph. As \( x \) increases, \( f(x) \) should approach the horizontal line \( y = \frac{1}{e} \). This suggests convergence towards \( \frac{1}{e} \), indicative of the function's behavior at infinity as per the limit definition of the exponential function.

Step 5: Calculate the Limit (Optional)

For further confirmation, consider calculating: \( \lim_{{x \to \infty}} \left( 1 - \frac{1}{x} \right)^x = \frac{1}{e} \). This algebraic approach matches our graphical observation, providing additional validation of \( f(x) \)'s behavior.

Key Concepts

Limit of a Function

In mathematics, the limit of a function describes the behavior of that function as the input approaches a certain point. Here, we're interested in the limit of the function \( f(x) = \left( 1 - \frac{1}{x} \right)^x \) as \( x \to \infty \). This involves understanding what value \( f(x) \) approaches as \( x \) becomes very large. For \( f(x) \), there's a well-known limit involving the natural number \( e \). As \( x \to \infty \), \( \left( 1 - \frac{1}{x} \right)^x \) approaches \( e^{-1} \). The mathematical representation is: \[ \lim_{{x \to \infty}} \left( 1 - \frac{1}{x} \right)^x = \frac{1}{e} \]This limit shows that as \( x \) increases, \( f(x) \) converges to \( \frac{1}{e} \), which is approximately 0.3679. Understanding this concept is crucial, as limits are foundational in calculus and help in analyzing function behaviors at boundaries. They form the basis for derivatives and integrals, which are the core operations of calculus.

Graphing Functions

Graphing functions is a valuable tool in visualizing and understanding mathematical concepts. For the function \( f(x) = \left( 1 - \frac{1}{x} \right)^x \), plotting gives us a clear representation of how \( f(x) \) behaves as \( x \) increases. To graph this function effectively, you need a range for \( x \) and a corresponding range for \( y \). From the original exercise, we're using \( x \) values from 0 to 20 and \( y \) values from 0 to 1. This specific range provides the best view of the function's approach towards the limiting value of \( \frac{1}{e} \). Additionally, it's useful to draw the line \( y = \frac{1}{e} \) on the same graph. This horizontal line acts as a reference, helping you see how \( f(x) \) moves closer to this value. Graphing the function alongside the reference line offers an immediate, visual way to interpret the function's trend and how it stabilizes around \( \frac{1}{e} \). With this visual analysis, the convergence and trend of the function become easier to understand compared to purely numerical or algebraic evaluation.

Convergence of Sequences

Convergence is a critical concept in both calculus and sequence analysis. When we talk about the convergence of sequences, we're referring to how a sequence of terms approaches a specific value as the sequence progresses towards infinity. For \( f(x) = \left( 1 - \frac{1}{x} \right)^x \), as \( x \) becomes larger, it forms a sequence that approaches a limit. Here, the sequence is the set of function values at various \( x \) values. With \( x \to \infty \), the function values converge to \( \frac{1}{e} \), demonstrating the convergence of the sequence defined by \( f(x) \).

• Convergence happens when the terms of a sequence get closer to a specific number, called the limit.
• For \( f(x) \), understanding convergence helps in predicting the behavior of this function at very large \( x \) values, without needing to calculate each successive term explicitly.
• In practical terms, this means that instead of evaluating \( f(x) \) at infinity, recognizing its convergence allows us to define its behavior with the limit \( \frac{1}{e} \).

Mastering the concept of convergence prepares students for deeper studies in analysis and enhances comprehension of how things like an entire function sequence can behave predictively and soundly as inputs grow indefinitely.