Problem 87 - Precalculus

Question

Plot the functions \(f(x)=\left(1+\frac{1}{x}\right)^{x}, g(x)=\left(1+\frac{2}{x}\right)^{x},\) and \(h(x)=\left(1+\frac{2}{x}\right)^{2 x}\) 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 \( x \to \infty \), \( f(x) \to e \) and \( g(x), h(x) \to e^2 \).

Step 1: Understand the Functions

First, let's look at the functions: \( f(x)=\left(1+\frac{1}{x}\right)^{x} \), \( g(x)=\left(1+\frac{2}{x}\right)^{x} \), and \( h(x)=\left(1+\frac{2}{x}\right)^{2x} \). As \( x \) approaches infinity, the terms \( \frac{1}{x} \) and \( \frac{2}{x} \) approach 0, simplifying the base of each expression.

Step 2: Determine Horizontal Asymptotes

Consider the limit of each function as \( x \to \infty \). For \( f(x) = \left(1+\frac{1}{x}\right)^{x} \), the limit is the mathematical constant \( e \), approximately 2.718. Similarly, for \( g(x) = \left(1+\frac{2}{x}\right)^{x} \), the asymptote is \( e^2 \). The function \( h(x) = \left(1+\frac{2}{x}\right)^{2x} \) also has \( e^2 \) as its limit. Thus, the horizontal asymptotes are \( e \) for \( f(x) \) and \( e^2 \) for both \( g(x) \) and \( h(x) \).

Step 3: Plot the Functions

Use graphing software or a graphing calculator to plot the functions. Review the continuity and behavior as \( x \) increases. Ensure that each function is plotted over a suitable range to visualize the horizontal asymptotes.

Step 4: Compare Function Values

Inspect the plots or calculate values for large \( x \). As predicted, \( f(x) \) will tend toward \( e \), while \( g(x) \) and \( h(x) \) will approach \( e^2 \). Compare the output of each function for increasing \( x \), confirming their relationship to the powers of \( e \).

Key Concepts

limits of functions

The concept of limits helps us understand what happens to a function's value as the variable approaches a certain point, often infinity. For example, consider the functions given in the original problem, such as \[ f(x)=\left(1+\frac{1}{x}\right)^x \] As \( x \) becomes larger, the term \( \frac{1}{x} \) shrinks towards zero, making the expression within the parentheses approach 1. Now, computing the limit: \[ \lim_{{x \to \infty}}\left(1+\frac{1}{x}\right)^x = e \] This means that as \( x \) approaches infinity, \( f(x) \) approaches the constant \( e \). Similar reasoning is used for the functions \( g(x) \) and \( h(x) \): - \( g(x)=\left(1+\frac{2}{x}\right)^x \): Approaches \( e^2 \) as \( x \to \infty \). - \( h(x)=\left(1+\frac{2}{x}\right)^{2x} \): Also tends towards \( e^2 \) because the exponent doubles as \( x \to \infty \). These limits indicate how the functions converge to these horizontal lines known as horizontal asymptotes.

plotting functions

To visually understand these functions and their limits, we plot them on a graph. Graphing can be achieved using software like Desmos or a scientific calculator.
Here's what to do while plotting:

• Select a broad range for \( x \). It's important to choose a range that effectively demonstrates the behavior as \( x \) gets large since we are interested in horizontal asymptotes.
• Aim to include negative and positive values of \( x \). However, focus more on the positive side where the asymptotic behavior is clearer.
• Notice how the functions behave as they approach their horizontal asymptotes. This will help in contrasting their long-term behavior.

By plotting, you can observe the following:- \( f(x) \) approaches a value close to \( e \) and settles along a horizontal line over large \( x \).- Both \( g(x) \) and \( h(x) \) approach the same horizontal line at \( e^2 \), drawing your attention on how exponentiation impacts their behavior compared to \( f(x) \). By viewing the graphs, the asymptotic behavior is clearer, making the relationship between functions and their limits more intuitive.

behavior at infinity

When we talk about a function's behavior at infinity, we mean how the functions descend or ascend as the variable \( x \) grows without bound. This aspect is crucial when considering functions like the ones in our exercise.
The behavior at infinity often results in horizontal asymptotes. A horizontal asymptote is a line that the graph of a function approaches but never touches as \( x \to \infty \) or \( x \to -\infty \).Consider the functions again:

• \( f(x) = \left(1+\frac{1}{x}\right)^x \): As \( x \to \infty \), the exponents reckon with the diminishing fractional components, guiding the function towards \( e \).
• \( g(x) = \left(1+\frac{2}{x}\right)^x \) and \( h(x) = \left(1+\frac{2}{x}\right)^{2x} \): Both functions aim towards \( e^2 \) due to their respective structures inferring double impact through a varying exponent or a squared influence.

The behavior in terms of impact and outcome for these equations tells us about stability and end behavior. They illustrate not only how math sees growth but the inherent predictability when scaling towards infinity. This enables us to better understand complex functions and anticipate their values.