Calculus: Early Transcendentals
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ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![### Rational Function Graph Analysis
The graph of a rational function \( f(x) = -\dfrac{(6x + 3)^2}{(x - 2)(x + 7)^2 (6x + 3)} \) is shown in the diagram above. This function has vertical and horizontal asymptotes, which are important in determining the function's behavior as \( x \) approaches specific values.
#### Graph Description:
- The graph exhibits a vertical asymptote at \( x = 2 \) and \( x = -7 \), where the function approaches \( +\infty \) or \( -\infty \).
- The graph has a horizontal asymptote at \( y = 0 \), indicating the function stabilizes horizontally as \( x \) approaches \( \pm \infty \).
#### Asymptotic and Limit Behavior:
To analyze the function \( f(x) \) at various points we must use limits:
1. **Limit as \( x \) approaches \( +\infty \)**:
\[
\lim_{{x \to +\infty}} f(x) =
\]
The value should be \( 0 \) because the degree of the numerator is less than the degree of the denominator.
2. **Limit as \( x \) approaches \( -\infty \)**:
\[
\lim_{{x \to -\infty}} f(x) =
\]
Similarly, the value should be \( 0 \).
3. **Limit as \( x \) approaches \( -7 \)**:
\[
\lim_{{x \to -7}} f(x) =
\]
The value should be \( \infty \) or \( -\infty \) due to the vertical asymptote at \( x = -7 \).
4. **Limit as \( x \) approaches \( -\dfrac{3}{6} \)**:
\[
\lim_{{x \to -\dfrac{3}{6}}} f(x) =
\]
Since \( -\dfrac{3}{6} \) simplifies to \( -0.5 \), which is not near any vertical asymptotes, this limit should exist and can be computed directly.
#### Fill in the Following Blanks:
- a. \( \](https://content.bartleby.com/qna-images/question/64eb814c-7ad8-4374-9b62-4531d74a96e9/2220959e-5b96-4f26-9cbd-e04ac8152e40/scw8nt4.png)
Transcribed Image Text:### Rational Function Graph Analysis
The graph of a rational function \( f(x) = -\dfrac{(6x + 3)^2}{(x - 2)(x + 7)^2 (6x + 3)} \) is shown in the diagram above. This function has vertical and horizontal asymptotes, which are important in determining the function's behavior as \( x \) approaches specific values.
#### Graph Description:
- The graph exhibits a vertical asymptote at \( x = 2 \) and \( x = -7 \), where the function approaches \( +\infty \) or \( -\infty \).
- The graph has a horizontal asymptote at \( y = 0 \), indicating the function stabilizes horizontally as \( x \) approaches \( \pm \infty \).
#### Asymptotic and Limit Behavior:
To analyze the function \( f(x) \) at various points we must use limits:
1. **Limit as \( x \) approaches \( +\infty \)**:
\[
\lim_{{x \to +\infty}} f(x) =
\]
The value should be \( 0 \) because the degree of the numerator is less than the degree of the denominator.
2. **Limit as \( x \) approaches \( -\infty \)**:
\[
\lim_{{x \to -\infty}} f(x) =
\]
Similarly, the value should be \( 0 \).
3. **Limit as \( x \) approaches \( -7 \)**:
\[
\lim_{{x \to -7}} f(x) =
\]
The value should be \( \infty \) or \( -\infty \) due to the vertical asymptote at \( x = -7 \).
4. **Limit as \( x \) approaches \( -\dfrac{3}{6} \)**:
\[
\lim_{{x \to -\dfrac{3}{6}}} f(x) =
\]
Since \( -\dfrac{3}{6} \) simplifies to \( -0.5 \), which is not near any vertical asymptotes, this limit should exist and can be computed directly.
#### Fill in the Following Blanks:
- a. \( \
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- If -a2 + 6x – 7 < f(x) < 2x – 3 for all r, find lim f(x). Cite any relevant theorems used.arrow_forwardEvaluate the following. 1. lim (2x – 11) x-4 x+4 2. lim x-2 l(x-1)(x+2),arrow_forwardSuppose that the graph of a function f is known. Explain how the graph of of y = f(x) – 2 differs from the graph of y = f(x – 2).arrow_forward
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