Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Use a table and/or graph to find the asymptote\((s)\) of each function. $$f(x)=\frac{e^{x}}{e^{x}-1}$$

Step-by-Step Solution

Answer

Vertical asymptote at \( x = 0 \), horizontal asymptote at \( y = 1 \).

Step 1: Identify the Function

The given function is \( f(x) = \frac{e^x}{e^x-1} \). This is a rational function where the numerator is \( e^x \) and the denominator is \( e^x - 1 \).

Step 2: Determine Vertical Asymptote

Vertical asymptotes occur when the denominator of a function is equal to zero. Set the denominator \( e^x - 1 \) equal to zero and solve: \( e^x - 1 = 0 \Rightarrow e^x = 1 \). Taking the natural logarithm of both sides, we find \( x = 0 \). Thus, there is a vertical asymptote at \( x = 0 \).

Step 3: Determine Horizontal Asymptote

To find the horizontal asymptote, analyze the behavior of \( f(x) \) as \( x \to \infty \) and \( x \to -\infty \). For large values of \( x \), both the numerator and the denominator \( e^x \) dominate, leading to the form \( \frac{e^x}{e^x} = 1 \). Therefore, the horizontal asymptote is \( y = 1 \).

Step 4: Use a Graph to Verify Asymptotes

Plot the function or use graphing technology to verify the asymptotes. The graph will show a vertical asymptote at \( x = 0 \) where the function approaches infinity on either side. The horizontal asymptote \( y = 1 \) will confirm that as \( x \to \infty \), the function approaches \( y = 1 \).

Step 5: Summarize the Asymptotes

The function \( f(x) = \frac{e^x}{e^x-1} \) has a vertical asymptote at \( x=0 \) and a horizontal asymptote at \( y=1 \).

Key Concepts

Understanding Rational Functions

A rational function is a type of function that is expressed as the ratio of two polynomials. In mathematical terms, it is written as \( f(x) = \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomial functions, and \( q(x) eq 0 \). For the exercise function \( f(x) = \frac{e^x}{e^x-1} \), the numerator is \( e^x \) and the denominator is \( e^x - 1 \). This function fits into the category of rational functions since it is essentially a ratio of two expressions involving exponentials, which can be considered as specialized polynomials for our analysis. By understanding that this is a rational function, we can predict that certain characteristics—such as asymptotes—will likely be present. Rational functions can often have breaks or discontinuities in their graph, where the function may approach infinity or flatten out as part of its overall behavior.

Exploring Vertical Asymptotes

Vertical asymptotes occur in rational functions when the denominator equals zero, causing the function to approach infinity at that point. In simpler terms, it's where the function "blows up" and becomes very large or very small. To find the vertical asymptote of our function \( f(x) = \frac{e^x}{e^x-1} \), we solve \( e^x - 1 = 0 \). This reduces to \( e^x = 1 \), or \( x = 0 \). Thus, the function has a vertical asymptote at \( x = 0 \).

What this means on the graph of the function is that as \( x \) gets very close to 0 from either positive or negative side, the value of \( f(x) \) becomes extremely large (approaching infinity) or extremely small (approaching negative infinity). The graph will never actually touch or cross the line \( x = 0 \), but will instead shoot upwards or downwards sharply. This feature is crucial when examining the behavior of rational functions.

Understanding Horizontal Asymptotes

Horizontal asymptotes reflect the values that the function approaches as \( x \) moves towards infinity or negative infinity. In the function \( f(x) = \frac{e^x}{e^x-1} \), we examine the behavior as \( x \to \,\pm \infty \). For large positive values of \( x \), both the numerator and the denominator become dominated by \( e^x \), which simplifies our expression to \( \frac{e^x}{e^x} = 1 \). Hence, there is a horizontal asymptote at \( y = 1 \).

This means that as we move further and further along the x-axis in the positive direction, the values of \( f(x) \) become indistinguishably close to 1. Similarly, for negative infinity, \( e^x \) approaches zero, simplifying the ratio to a form that approaches 1 as well. Horizontal asymptotes indicate that despite any initial fluctuations or rapid changes, the function will tend to level off to these specific values far along the x-axis.

Graphical Analysis of Asymptotes

Graphical analysis is a powerful tool in understanding the characteristics of rational functions. By visually plotting \( f(x) = \frac{e^x}{e^x-1} \), we can observe the behavior mentioned previously, such as the vertical and horizontal asymptotes. A graph will show the vertical asymptote at \( x = 0 \), where the function diverges sharply, suggesting the points where the function values are hugely positive or negative.

Furthermore, the graph will illustrate the flattening behavior as \( x \to \infty \), confirming the horizontal asymptote \( y = 1 \). By seeing this on a graph, it becomes evident how the function approaches and more importantly, never reaches, these asymptotic lines. Graphical analysis not only verifies our analytical predictions about the asymptotes but also provides a practical visual representation of the overall dynamics of the function, making it easier to understand complex concepts at a glance.