Problem 81 - Precalculus with Limits

Question

Graphical Analysis Use a graphing utility to graph \(y_{1}=(1+1 / x)^{x}\) and \(y_{2}=e\) in the same viewing window. Using the trace feature, explain what happens to the graph of \(y_{1}\) as \(x\) increases.

Step-by-Step Solution

Answer

The graph of \(y_{1}=(1+1 / x)^{x}\) approaches the horizontal line \(y = e\) as \(x\) increases, i.e., \(y_{1}\) is asymptotic to the line \(y = e\).

Step 1: Graph functions

Start by graphing the two functions \(y_{1}=(1+1 / x)^{x}\) and \(y_{2}=e\) in a same window using a graphing utility. Observe the general shape of the functions and their intersections.

Step 2: Inspection of \( y_1 \) as \( x \) increases

Use the trace feature to navigate the values of the function \( y_1 \) for larger and larger \( x \) values. Note the value of \( y_1 \) as \( x \) grows without bound. You'll observe the function's values get closer and closer to some fixed value as \( x \) increases.

Step 3: Deduce the behavior of \( y_1 \) as \( x \) approaches infinity

Through your observations, you can realise that the \( y_1 \) graph is approaching the horizontal line \( y = e \) as \( x \) gets larger. This means that the function \( y_1 \) is asymptotic to the line \( y = e \), or in other words, the limit of \( (1+1 / x)^{x} \) as \( x \) approaches infinity is \( e \).

Key Concepts

Asymptotic Behavior

Understanding the asymptotic behavior of functions is a critical aspect of graphical analysis in precalculus. Asymptotic behavior describes how a function behaves as the input values either grow very large (\( x \to \infty \)) or very small (\( x \to -\infty \)). Specifically, we are interested in whether the function approaches a particular line that is called an asymptote.
An asymptote can be a horizontal, vertical, or slanted line that the graph of the function approaches but never reaches. The concept is helpful to predict the end behavior of the graph and understand the long-term trends of the function's values. For example, in the exercise, the function \( y_1 = (1 + 1/x)^x \) approaches the horizontal line \( y = e \) as \( x \) increases, which indicates that \( y = e \) serves as a horizontal asymptote.

Limits

The concept of limits forms the foundation of understanding continuous behavior in functions, especially when considering inputs approaching infinity or a specific value. In simple terms, a limit is the value that a function or sequence 'approaches' as the input (or index) approaches some value.
When you hear 'the limit of \( f(x) \) as \( x \) approaches \( a \) is \( L \)', it means that as \( x \) gets closer and closer to \( a \), the values of \( f(x) \) get arbitrarily close to \( L \), symbolized as \( \lim_{x\to a} f(x) = L \). In the context of the provided exercise, we're examining the limit of the function \( (1+1/x)^x \) as \( x \) approaches infinity, where the value approached is the constant \( e \).

Graphing Utilities

Graphing utilities are invaluable tools in precalculus that allow for visual representation and exploration of mathematical functions. These digital tools provide a dynamic environment to plot functions, analyze their behavior, and comprehend complex concepts visually—turning abstract ideas into something concrete.
Features like zoom, trace, and table view enable students to carry out a detailed analysis. For instance, by employing the trace feature, one can observe how the values of a function change with the input, offering an interactive learning experience. In our exercise, a graphing utility helps to illustrate how \( y_1 = (1 + 1/x)^x \) behaves as \( x \) grows, providing visual evidence of the limit and asymptotic behavior.

Exponential Functions

Exponential functions, such as \( y = b^x \), where \( b \) is a positive constant, are dramatically affected by changes in the value of \( x \). These functions grow or decay at rates proportional to their current value, leading to graphs that can quickly soar upwards or downwards. The base of the exponential function determines its growth rate—larger bases grow more rapidly.
In the context of the exercise, the function \( y_1 = (1 + 1/x)^x \) is particularly interesting because as \( x \) increases, the expression \( 1 + 1/x \) approaches 1, but the exponent \( x \) grows. This competition between the base approaching 1 and the exponent increasing indefinitely causes the function to approach the unique number \( e \), known as Euler's number, which itself is the base of the natural exponential function \( e^x \).