Problem 59 - Algebra and Trigonometry

Question

58–59 ? Graph the function and comment on vertical and horizontal asymptotes. $$ y=\frac{e^{x}}{x} $$

Step-by-Step Solution

Answer

Vertical asymptote at \( x = 0 \) and horizontal asymptote at \( y = 0 \) as \( x \to -\infty \).

Step 1: Understand the Function

We are asked to analyze the function \( y = \frac{e^x}{x} \). This is a rational function, where the numerator is the exponential function \( e^x \) and the denominator is the linear function \( x \).

Step 2: Identify Vertical Asymptotes

Vertical asymptotes occur where the denominator equals zero and the numerator is non-zero. Here, \( x = 0 \) could be a candidate, but the numerator \( e^x \) does not equal zero. Therefore, \( x = 0 \) is a vertical asymptote.

Step 3: Consider Horizontal Asymptotes

Horizontal asymptotes relate to the end behavior of the function as \( x \to \pm \infty \). For \( y = \frac{e^x}{x} \), as \( x \to \infty \), \( e^x \) grows much faster than \( x \), so the function does not approach a constant value, indicating no horizontal asymptote at \( x \to \infty \). However, as \( x \to -\infty \), the function approaches zero because \( e^x \) decays faster than the linear growth of the negative \( x \). Thus, there is a horizontal asymptote at \( y = 0 \).

Step 4: Graph the Function

To graph \( y = \frac{e^x}{x} \), note that for \( x > 0 \), the graph increases, and for \( x < 0 \), the graph decreases and approaches zero. Plotting several points can help visualize this, but keep the asymptotic behavior in mind.

Step 5: Summarize Asymptotes

From the analysis above, \( x = 0 \) is a vertical asymptote. The function approaches \( y = 0 \) as \( x \to -\infty \), indicating a horizontal asymptote. However, as \( x \to \infty \), there is no horizontal asymptote.

Key Concepts

Rational Functions

A rational function is expressed as the ratio of two polynomials. In the case of our exercise, the function is given by \( y = \frac{e^x}{x} \). Here, the numerator is the exponential function \( e^x \) and the denominator is a linear function, \( x \).

This function is interesting because it mixes the properties of exponential and polynomial functions.

• The numerator, \( e^x \), grows rapidly as \( x \) increases. This rapid growth affects the behavior of the function significantly.
• The denominator \( x \) affects the function by determining points where the function is undefined (specifically, where \( x = 0 \)).

Understanding rational functions includes paying attention to where they are undefined, as these are potential vertical asymptotes, and considering the degree of the polynomials involved to study their behavior at the extremes of the number line.

Exponential Functions

Exponential functions involve a constant base raised to a variable exponent, in this case, the base \( e \) raised to the power of \( x \), denoted as \( e^x \).

This function grows extremely fast as \( x \) becomes large, significantly influencing the behavior of any function of which it is a part. This rapid growth is evident in our rational function example, where the exponential term in the numerator largely determines the behavior of the entire function at large values of \( x \).

• The base \( e \) is known for its natural growth properties, making such expressions common in calculus and mathematical modeling.
• In particular, as \( x \to \infty \), the value of \( e^x \) increases so rapidly that it outpaces any polynomial denominator (like \( x \) in our function).
• Conversely, as \( x \to -\infty \), \( e^x \) rapidly approaches zero. This behavior is crucial for determining asymptotic properties.

Graphing Functions

Graphing functions like \( y = \frac{e^x}{x} \) involves understanding its asymptotes and growth behavior across different regions of \( x \).

Graphing helps visualize how the function behaves relatively to its asymptotes:

• Vertical asymptote at \( x = 0 \): This is where the function is undefined, dividing the graph into two distinct sections, one for positive \( x \) and one for negative \( x \).
• Behavior for \( x > 0 \): The graph increases as \( x \) does because \( e^x \) grows faster than \( x \).
• Behavior for \( x < 0 \): The function decreases and approaches zero. This is because the negative effect of \( x \) in the denominator reduces the value of the entire function, despite the exponential term eventually tending towards zero.

Graphing can guide us to see how these mathematical properties visually come together, shedding light on the overall nature of the function.

End Behavior

Understanding a function's end behavior reveals what happens to the function’s value as \( x \) approaches infinity or negative infinity. In our function, \( y = \frac{e^x}{x} \), analyzing end behavior helps identify asymptotic trends.

For \( y = \frac{e^x}{x} \), consider:

• As \( x \to \infty \): The numerator \( e^x \) grows exponentially, and much faster than the linear growth of \( x \). Therefore, there is no horizontal asymptote, as the function does not approach a specific value.
• As \( x \to -\infty \): The function approaches zero. This happens because the value of \( e^x \) decreases much faster than \( x \) becomes negatively large, suggesting a horizontal asymptote at \( y = 0 \).

Recognizing these patterns allows predictions about function values at extreme ranges, crucial for understanding and graphing functions.