(6x + 3)² (x – 2)(x + 7)°(6x + 3) does not exist, "" if the function output increases without bound, or – o if the function output decreases without bound. The graph of a rational function f(x) = is shown above. Fill in the following blanks. Write "DNE" if the limit a. lim f(x) = Preview b. lim f(x) = Preview c. lim f(x) = Preview x→-7 d. lim f(x) = x→-3/6 Preview
(6x + 3)² (x – 2)(x + 7)°(6x + 3) does not exist, "" if the function output increases without bound, or – o if the function output decreases without bound. The graph of a rational function f(x) = is shown above. Fill in the following blanks. Write "DNE" if the limit a. lim f(x) = Preview b. lim f(x) = Preview c. lim f(x) = Preview x→-7 d. lim f(x) = x→-3/6 Preview
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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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. \( \](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F64eb814c-7ad8-4374-9b62-4531d74a96e9%2F2220959e-5b96-4f26-9cbd-e04ac8152e40%2Fscw8nt4.png&w=3840&q=75)
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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