Problem 20 - 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: Understand the function

We start with the given function \( f(x) = \frac{e^x}{e^x - 1} \). To find the asymptotes, we need to understand where the function either does not exist or heads towards infinity. This typically involves finding points where the denominator is zero (for vertical asymptotes) and understanding the behavior of the function as \( x \to \pm \infty \) (for horizontal or oblique asymptotes).

Step 2: Determine potential vertical asymptotes

To find vertical asymptotes, set the denominator equal to zero: \( e^x - 1 = 0 \). Solving this gives \( e^x = 1 \), hence \( x = 0 \). This means there is a vertical asymptote at \( x = 0 \).

Step 3: Analyze behavior as \(x\to\infty\)

To find horizontal asymptotes, observe the behavior as \( x \) approaches infinity. For large \( x \), the terms \( e^x \) in the numerator and the denominator are dominant, simplifying the function to \( \frac{e^x}{e^x - 1} \approx \frac{e^x}{e^x} = 1 \). Therefore, as \( x \to \infty \), \( f(x) \to 1 \). Thus, \( y = 1 \) is a horizontal asymptote.

Step 4: Analyze behavior as \(x\to - infty\)

For \( x \to -\infty \), \( e^x \to 0 \), so the function becomes \( \frac{0}{0 - 1} = 0 \). Therefore, as \( x \to -\infty \), \( f(x) \to 0 \). Thus, \( y = 0 \) is not an asymptote as it's not approached at finite limits, but an interesting behavior.

Step 5: Summary of Asymptotes

Vertical asymptote at \( x = 0 \). Horizontal asymptote at \( y = 1 \). No other asymptotes are present based on the behavior of the function as \( x \to \pm\infty \).

Key Concepts

Vertical Asymptote

When dealing with functions, vertical asymptotes are lines that a graph approaches but never touches or crosses. These typically occur where the function becomes undefined, which is usually identified by setting the denominator of a rational function to zero. In the function \( f(x) = \frac{e^x}{e^x - 1} \), finding the vertical asymptote involves solving \( e^x - 1 = 0 \). Solving this equation gives \( e^x = 1 \) which simplifies to \( x = 0 \). Hence, there is a vertical asymptote at \( x = 0 \).
Vertical asymptotes are significant in indicating where the graph tends towards positive or negative infinity, turning abruptly as it approaches the line. Just remember: the function does not actually touch or cross the asymptote! Elements like these can often make a graph appear quite dramatic.

Horizontal Asymptote

Horizontal asymptotes show the behavior of a function as \( x \) increases or decreases without bound. They tell us the value the function is approaching, providing insight into the end behavior of the graph. In our function \( f(x) = \frac{e^x}{e^x - 1} \), as \( x \to \infty \), both the numerator and denominator grow similarly, simplifying our function to approximately \( 1 \). Thus, the horizontal asymptote for this function is \( y = 1 \).
Horizontal asymptotes give a sense of stability in the graph, revealing where the function tends to level out as \( x \) stretches towards infinity. It’s like the function is saying: "I’ll get really close to this continuous line…I promise!"

Behavior of Exponential Functions

Exponential functions, such as those including \( e^x \), grow rapidly either upwards or downwards, making them fascinating and powerful. The term \( e^x \) is significant because as \( x \to \, \infty \) or \( x \to -\infty \), the impacts vary drastically:

• For \( x \to \infty \), it blows up, making the dominance in expressions like \( \frac{e^x}{e^x - 1} \) evident.
• For \( x \to -\infty \), \( e^x \to 0 \), leading to simplifications like \( \frac{0}{0-1} = 0 \).

Understanding these behaviors allows you to predict and analyze the dramatic changes in the graph, pinpointing where it grows steeply or flattens out.

Graphical Analysis

Graphical analysis involves visually interpreting the behavior of functions using graphs. Seeing the actual picture can often make understanding asymptotes and function behavior much easier! For \( f(x) = \frac{e^x}{e^x - 1} \), you’ll notice that

• Near \( x = 0 \), the curve will approach the vertical line but never touch it due to the asymptote.
• As \( x \to \infty \), the graph levels off near \( y = 1 \), making it evident as a horizontal asymptote.

Graphical analysis proves useful for identifying asymptotic behavior at a glance, providing the bigger picture beyond algebraic manipulation. Simply put, the graph speaks a thousand words.

Limits

Limits help us understand how a function behaves near certain points or as \( x \) approaches infinity or negative infinity. They are fundamental in defining asymptotes. For the function \( f(x) = \frac{e^x}{e^x - 1} \):

• The limit as \( x \to 0 \) from the right shows the behavior towards the vertical asymptote.
• The limits \( x \to \infty \) and \( x \to -\infty \) demonstrate the approach towards potential horizontal asymptotes.

Limits don't just account for the presence of asymptotes but explain behavior in terms of proximity and convergence towards a line or a point. They offer a structured way to predict the unending story of a function’s journey.