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-8 -6 АУ 6 41 d N -41 2 4 X

-8 -6 АУ 6 41 d N -41 2 4 X

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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find the vertical asymptote
This graph illustrates a rational function with an asymptote and a highlighted point. Here's a detailed description to help understand the graph better: ### Graph Description: 1. **Axes and Grid**: - The graph is plotted on a Cartesian coordinate system with a grid. - The x-axis and y-axis are labeled from -8 to 8. - The x-axis extends horizontally, and the y-axis extends vertically. 2. **Asymptote**: - A dashed vertical red line at \( x = -2 \) indicates the presence of a vertical asymptote. - This suggests that the function approaches infinity as \( x \) approaches -2 from the left and negative infinity as \( x \) approaches -2 from the right. 3. **Function Behavior**: - The graph of the function is drawn in red. - For \( x \) less than -2, the function decreases rapidly as it approaches the vertical asymptote from the left. As x approaches \( -2 \) from the left, the y-value approaches positive infinity. - For \( x \) between -2 and positive infinity, the graph increases from negative infinity at \( x = -2 \), crosses the y-axis, passes through an indicated point on the graph, and extends towards positive infinity as \( x \) grows larger. 4. **Highlighted Point**: - A red dot is placed on the graph at the coordinates \( (2, 1) \). ### Key Notes: - Understanding the behavior of the function near the asymptote is crucial. It affects the domain of the function, excluding x = -2 because the function tends to infinity or negative infinity at this point. - The horizontal extension of the graph on both sides shows the end behavior, indicating how the function behaves as \( x \) approaches positive and negative infinity. - The red dot indicates a specific value of the function at \( x = 2 \), which is \( y = 1 \). This graph helps visualize the relationship between \( x \) and \( y \) for a given rational function, illustrating how it behaves near critical points and far from the origin.
Transcribed Image Text:This graph illustrates a rational function with an asymptote and a highlighted point. Here's a detailed description to help understand the graph better: ### Graph Description: 1. **Axes and Grid**: - The graph is plotted on a Cartesian coordinate system with a grid. - The x-axis and y-axis are labeled from -8 to 8. - The x-axis extends horizontally, and the y-axis extends vertically. 2. **Asymptote**: - A dashed vertical red line at \( x = -2 \) indicates the presence of a vertical asymptote. - This suggests that the function approaches infinity as \( x \) approaches -2 from the left and negative infinity as \( x \) approaches -2 from the right. 3. **Function Behavior**: - The graph of the function is drawn in red. - For \( x \) less than -2, the function decreases rapidly as it approaches the vertical asymptote from the left. As x approaches \( -2 \) from the left, the y-value approaches positive infinity. - For \( x \) between -2 and positive infinity, the graph increases from negative infinity at \( x = -2 \), crosses the y-axis, passes through an indicated point on the graph, and extends towards positive infinity as \( x \) grows larger. 4. **Highlighted Point**: - A red dot is placed on the graph at the coordinates \( (2, 1) \). ### Key Notes: - Understanding the behavior of the function near the asymptote is crucial. It affects the domain of the function, excluding x = -2 because the function tends to infinity or negative infinity at this point. - The horizontal extension of the graph on both sides shows the end behavior, indicating how the function behaves as \( x \) approaches positive and negative infinity. - The red dot indicates a specific value of the function at \( x = 2 \), which is \( y = 1 \). This graph helps visualize the relationship between \( x \) and \( y \) for a given rational function, illustrating how it behaves near critical points and far from the origin.
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