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Magazine Chapter 5: Problem 59
58鈥�59 ? Graph the function and comment on vertical and horizontal asymptotes. $$ y=\frac{e^{x}}{x} $$
Short Answer
Expert verified
Vertical asymptote at \( x = 0 \) and horizontal asymptote at \( y = 0 \) as \( x \to -\infty \).
Step by step solution
01
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 \( x \).
02
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.
03
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 \( y = 0 \).
04
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.
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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 \( x = 0 \)).
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 \( x \).
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:
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.
End Behavior
Understanding a function's end behavior reveals what happens to the function鈥檚 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:
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 \( y = 0 \).
