### Rational Function Graph AnalysisThe 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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(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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Concept explainers

Limits

When it comes to calculus, limits are considered to be a very important topic of discussion. The main importance of limit lies in expressing the value of a function as it gets closer to a certain value. Also, limits can help you to understand other topics, such as continuity and differentiability better. In a broader sense, every calculus notion is a limit. Other branches of calculus have also been derived from limits.

Question

### 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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