Problem 88 - Precalculus

Question

Plot the functions \(f(x)=\left(1+\frac{1}{x}\right)^{x}, g(x)=\left(1-\frac{1}{x}\right)^{x},\) and \(h(x)=\left(1-\frac{2}{x}\right)^{x}\) in the same viewing screen. Compare their horizontal asymptotes as \(x\) increases. What can you say about the function values of \(f, g,\) and \(h\) in terms of the powers of \(e\) as \(x\) increases?

Step-by-Step Solution

Answer

As \(x\) increases, \(f(x)\to e\), \(g(x)\to \frac{1}{e}\), and \(h(x)\to \frac{1}{e^2}\).

Step 1: Understanding the Functions

First, let's identify the given functions:- \( f(x) = \left(1 + \frac{1}{x}\right)^x \)- \( g(x) = \left(1 - \frac{1}{x}\right)^x \)- \( h(x) = \left(1 - \frac{2}{x}\right)^x \)These expressions are limits typically seen in calculus related to the base of the natural logarithm, \(e\).

Step 2: Recognizing Asymptotic Behavior

As \(x\) approaches infinity, the functions are related to exponential forms:- For \( f(x) \), \( \left(1 + \frac{1}{x}\right)^x \to e \) as \(x \to \infty\).- For \( g(x) \), \( \left(1 - \frac{1}{x}\right)^x \to \frac{1}{e} \) as \(x \to \infty\).- For \( h(x) \), \( \left(1 - \frac{2}{x}\right)^x \to \frac{1}{e^2} \) as \(x \to \infty\).

Step 3: Developing Graphical Representation

Plot each function over a reasonable domain to observe their graph behaviors:- Choose a large domain for \(x\), such as \(x = 1\) to \(x = 1000\).- Create a graph and plot \(f(x)\), \(g(x)\), and \(h(x)\) on the same screen.- Use software or tools like Desmos, MATLAB, or a graphing calculator.

Step 4: Analyze Graphs for Horizontal Asymptotes

By examining the graphs:- \(f(x)\) will show a horizontal asymptote approaching \(e\).- \(g(x)\) will approach a horizontal asymptote of \(\frac{1}{e}\).- \(h(x)\) will approach a horizontal asymptote of \(\frac{1}{e^2}\).

Step 5: Conclusion on the Function Values

As \(x\) increases, it is evident that:- \(f(x)\) approaches \(e\),- \(g(x)\) approaches \(e^{-1}\),- \(h(x)\) approaches \(e^{-2}\).These trends indicate each function’s values tend to powers of \(e\), showcasing the nature of limits in exponential forms.

Key Concepts

Horizontal Asymptotes

In calculus, a horizontal asymptote tells us about the behavior of a function as it goes toward infinity or negative infinity. It is a constant streamline that the function approaches as it diverges. When exploring the functions from our exercise, each has its own horizontal asymptote:

• For the function \( f(x) = \left(1 + \frac{1}{x}\right)^x \), as \(x\) approaches infinity, the function approaches a horizontal asymptote of \( e \). This is because it represents the classic limit definition of the natural exponential function, reflecting the very base of the natural logarithm.
• In the case of \( g(x) = \left(1 - \frac{1}{x}\right)^x \), the horizontal asymptote is \( \frac{1}{e} \), as \(x\) tends towards infinity, approaching the reciprocal of the natural exponential base.
• Lastly, the function \( h(x) = \left(1 - \frac{2}{x}\right)^x \) asymptotically approaches \( \frac{1}{e^2} \). Here, the function relates directly to exponential decay with its limit.

Understanding these asymptotes helps in predicting the long-term behavior of functions in calculus, especially crucial when considering real-world applications.

Exponential Functions

Exponential functions are foundational in calculus, defined by constants raised to variable powers. The functions considered in our problem showcase the effects of limits within exponential expressions involving the base of the natural logarithm, \(e\).

• For \( f(x) \), the expression \( \left(1 + \frac{1}{x}\right)^x \) is a standard representation leading to the number \(e\), a transcendental number central to many growth and decay processes in nature.
• The function \( g(x) = \left(1 - \frac{1}{x}\right)^x \) helps demonstrate the nature of exponential decay as it approaches the value \( \frac{1}{e} \).
• In a similar fashion, \( h(x) = \left(1 - \frac{2}{x}\right)^x \) embodies exponential decay much steeper, indicated by approaching \( \frac{1}{e^2} \).

These functions effectively illustrate how exponential limits are applied in calculus, with \(e\) being particularly significant in representing complex relationships in various real-world scenarios including biology, finance, and physics.

Graphical Analysis

Graphical analysis is a powerful means of understanding the behavior of functions visually. By plotting \( f(x) \), \( g(x) \), and \( h(x) \) on a large domain, such as from \(x=1\) to \(x=1000\), we can directly observe their convergence to horizontal asymptotes.

• \(f(x)\) shows a curve rising towards the line \(y = e\) as \(x\) increases, confirming its predicted asymptotic behavior.
• On the other hand, \(g(x)\) declines towards \(y = \frac{1}{e}\), highlighting a stable approach to this lower exponential value.
• Finally, \(h(x)\) exhibits an even sharper decline toward \(y = \frac{1}{e^2}\), displaying the deeper exponential decay as derived theoretically.

Graphing such functions emphasizes the convergence of these functions to specific values, reflecting their exponential nature. It enhances comprehension by linking algebraic limits to graphical insights, pivotal for students to grasp the concept of limits concretely.