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3 1 -4 -2 4 -1 -2 2. 3.

3 1 -4 -2 4 -1 -2 2. 3.

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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I don't believe this has a Horizontal Asymptote but I am not sure.

This graph displays the function \( f(x) = \frac{1}{x^2 - 1} \). ### Graph Breakdown: **Axes:** - The x-axis ranges from approximately -4.5 to 6.5. - The y-axis ranges from approximately -3.5 to 3.5. **Function Details:** - The graph of the function is shown in red. - The function has vertical asymptotes at \( x = -1 \) and \( x = 1 \), represented by blue dashed lines. - As \( x \) approaches -1 and 1 from the left and right, the function approaches positive or negative infinity, which is typical behavior around vertical asymptotes. **Behavior of the Function:** - In the intervals \( (-\infty, -1) \) and \( (1, \infty) \), the function decreases towards zero as \( x \) moves away from the asymptotes. - In the interval \( (-1, 1) \), the function transitions from negative infinity to positive infinity, indicating a change in sign as it passes through these asymptotes. This visualization helps in understanding rational functions, specifically how vertical asymptotes affect the behavior of such functions. Vertical asymptotes occur in rational functions where the denominator equals zero and the numerator is non-zero.
Transcribed Image Text:This graph displays the function \( f(x) = \frac{1}{x^2 - 1} \). ### Graph Breakdown: **Axes:** - The x-axis ranges from approximately -4.5 to 6.5. - The y-axis ranges from approximately -3.5 to 3.5. **Function Details:** - The graph of the function is shown in red. - The function has vertical asymptotes at \( x = -1 \) and \( x = 1 \), represented by blue dashed lines. - As \( x \) approaches -1 and 1 from the left and right, the function approaches positive or negative infinity, which is typical behavior around vertical asymptotes. **Behavior of the Function:** - In the intervals \( (-\infty, -1) \) and \( (1, \infty) \), the function decreases towards zero as \( x \) moves away from the asymptotes. - In the interval \( (-1, 1) \), the function transitions from negative infinity to positive infinity, indicating a change in sign as it passes through these asymptotes. This visualization helps in understanding rational functions, specifically how vertical asymptotes affect the behavior of such functions. Vertical asymptotes occur in rational functions where the denominator equals zero and the numerator is non-zero.
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