Use the graph of the rational function to complete the following statement. 40 As x 3*, f(x)→ As x → 3*, f(x) →. to to Asymptotes are shown as dashed lines. The horizontal asymptote is y = - 3. The vertical asymptotes are x = -3 and x = 3.
Use the graph of the rational function to complete the following statement. 40 As x 3*, f(x)→ As x → 3*, f(x) →. to to Asymptotes are shown as dashed lines. The horizontal asymptote is y = - 3. The vertical asymptotes are x = -3 and x = 3.
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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![### Understanding Rational Functions through Graphs
**Instructions:**
Use the graph of the rational function to complete the following statement.
**Given:**
1. \( \text{As } x \rightarrow 3^+ \), \( f(x) \rightarrow \_\_\_\_\_\_\_\_ \)
2. \( \text{As } x \rightarrow 3^- \), \( f(x) \rightarrow \_\_\_\_\_\_\_\_ \)
**Graph Details:**
The graph shown is a plot of a rational function. It includes vertical and horizontal asymptotes to guide your understanding of the function's limits and behavior.
- **Graph Explanation:**
The graph is a plot with both vertical and horizontal axes ranging approximately from -10 to 10. The rational function is indicated by a smooth curve in blue color. Key features of the graph include three asymptotes:
- **Horizontal Asymptote:** \( y = -3 \)
- **Vertical Asymptotes:** \( x = -3 \) and \( x = 3 \)
Asymptotes are shown as dashed lines. The purple curves on the graph approach the asymptotes but never touch them, indicating how the function behaves as it nears these critical values.
**Function Behavior Analysis:**
To complete the given statements:
1. To determine the behavior of \( f(x) \) as \( x \) approaches \( 3 \) from the right (\( 3^+ \)):
- The graph shows that as \( x \) nears 3 from the right, the function \( f(x) \) trends towards positive infinity.
Therefore,
\[ \text{As } x \rightarrow 3^+ \), \( f(x) \rightarrow \infty \]
2. To determine the behavior of \( f(x) \) as \( x \) approaches \( 3 \) from the left (\( 3^- \)):
- The graph clearly depicts that as \( x \) nears 3 from the left, the function \( f(x) \) trends towards negative infinity.
Therefore,
\[ \text{As } x \rightarrow 3^- \), \( f(x) \rightarrow -\infty \]
By analyzing these trends, one can understand the behavior of the rational function as it approaches its vertical asymptotes.
### Conclusion:
Through the graphical representation of the rational function and its](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47cd433c-af7e-4add-b878-a95e599142fb%2F1eaa93d4-63be-4432-b0d6-7d9b93bf22c1%2F1tdnqhc_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding Rational Functions through Graphs
**Instructions:**
Use the graph of the rational function to complete the following statement.
**Given:**
1. \( \text{As } x \rightarrow 3^+ \), \( f(x) \rightarrow \_\_\_\_\_\_\_\_ \)
2. \( \text{As } x \rightarrow 3^- \), \( f(x) \rightarrow \_\_\_\_\_\_\_\_ \)
**Graph Details:**
The graph shown is a plot of a rational function. It includes vertical and horizontal asymptotes to guide your understanding of the function's limits and behavior.
- **Graph Explanation:**
The graph is a plot with both vertical and horizontal axes ranging approximately from -10 to 10. The rational function is indicated by a smooth curve in blue color. Key features of the graph include three asymptotes:
- **Horizontal Asymptote:** \( y = -3 \)
- **Vertical Asymptotes:** \( x = -3 \) and \( x = 3 \)
Asymptotes are shown as dashed lines. The purple curves on the graph approach the asymptotes but never touch them, indicating how the function behaves as it nears these critical values.
**Function Behavior Analysis:**
To complete the given statements:
1. To determine the behavior of \( f(x) \) as \( x \) approaches \( 3 \) from the right (\( 3^+ \)):
- The graph shows that as \( x \) nears 3 from the right, the function \( f(x) \) trends towards positive infinity.
Therefore,
\[ \text{As } x \rightarrow 3^+ \), \( f(x) \rightarrow \infty \]
2. To determine the behavior of \( f(x) \) as \( x \) approaches \( 3 \) from the left (\( 3^- \)):
- The graph clearly depicts that as \( x \) nears 3 from the left, the function \( f(x) \) trends towards negative infinity.
Therefore,
\[ \text{As } x \rightarrow 3^- \), \( f(x) \rightarrow -\infty \]
By analyzing these trends, one can understand the behavior of the rational function as it approaches its vertical asymptotes.
### Conclusion:
Through the graphical representation of the rational function and its
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