Find the Vertical and Horizontal asymptotes. Display answer in interval notation. The image illustrates the graph of a rational function. The grid is marked with horizontal and vertical lines representing units.**Graph Details:**- **Axes:** The graph features an $ x $-axis and a $ y $-axis. The $ x $-axis is labeled and extends horizontally, while the $ y $-axis is labeled and extends vertically. - **Curve Description:** - The curve is composed of two distinct branches, each located in different quadrants of the graph. - The left branch approaches the vertical asymptote from the left and the horizontal asymptote from below. - The right branch approaches the vertical asymptote from above and tends towards negative infinity along the horizontal asymptote. - **Asymptotes:** - **Vertical Asymptote:** A dashed line at $ x = 2 $, indicating that the function approaches infinity or negative infinity as $ x $ approaches 2 from the left and right. - **Horizontal Asymptote:** A dashed line at $ y = 1 $, indicating that as $ x $ moves towards positive or negative infinity, the function value approaches 1.The function displays typical characteristics of rational functions with asymptotic behavior, showcasing vertical and horizontal asymptotes as boundaries that the graph approaches but never crosses. The graph's shape and asymptotes suggest it could be of the form $ y = \frac{1}{x - 2} + 1 $ or a similar variation, emphasizing the function’s undefined value at $ x = 2 $ and horizontal leveling towards $ y = 1 $.

Answered: Find the Vertical and Horizontal…

Math

Advanced Math

Find the Vertical and Horizontal asymptotes.

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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Question

Find the Vertical and Horizontal asymptotes. Display answer in interval notation.

The image illustrates the graph of a rational function. The grid is marked with horizontal and vertical lines representing units.

**Graph Details:**

- **Axes:** The graph features an $ x $-axis and a $ y $-axis. The $ x $-axis is labeled and extends horizontally, while the $ y $-axis is labeled and extends vertically.

- **Curve Description:**
- The curve is composed of two distinct branches, each located in different quadrants of the graph.
- The left branch approaches the vertical asymptote from the left and the horizontal asymptote from below.
- The right branch approaches the vertical asymptote from above and tends towards negative infinity along the horizontal asymptote.

- **Asymptotes:**
- **Vertical Asymptote:** A dashed line at $ x = 2 $, indicating that the function approaches infinity or negative infinity as $ x $ approaches 2 from the left and right.
- **Horizontal Asymptote:** A dashed line at $ y = 1 $, indicating that as $ x $ moves towards positive or negative infinity, the function value approaches 1.

The function displays typical characteristics of rational functions with asymptotic behavior, showcasing vertical and horizontal asymptotes as boundaries that the graph approaches but never crosses. The graph's shape and asymptotes suggest it could be of the form $ y = \frac{1}{x - 2} + 1 $ or a similar variation, emphasizing the function’s undefined value at $ x = 2 $ and horizontal leveling towards $ y = 1 $.

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