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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please show the solution for this question - the answer for all three parts is negative infiniy thank you!
![### Limit Evaluation Problem
**Problem Statement:**
45. Use the function \( f \) below to determine each limit:
\[ f(x) = \frac{-2x}{(x+3)^2} \]
**Limits to be evaluated:**
A. \[ \lim_{{x \to -3^+}} f(x) \]
B. \[ \lim_{{x \to -3^-}} f(x) \]
C. \[ \lim_{{x \to -3}} f(x) \]
---
**Explanation:**
Given the function \( f(x) = \frac{-2x}{(x+3)^2} \), we need to find the limits of the function as \( x \) approaches -3 from the right (\( -3^+ \)), from the left (\( -3^- \)), and as \( x \) approaches -3 in general.
1. **Limit from the Right (\( x \to -3^+ \))**:
- Observe the behavior of the function as \( x \) gets closer to -3 from values greater than -3.
2. **Limit from the Left (\( x \to -3^- \))**:
- Observe the behavior of the function as \( x \) gets closer to -3 from values less than -3.
3. **Two-Sided Limit (\( x \to -3 \))**:
- Determine whether the right-hand limit and left-hand limit are equal and thus conclude the two-sided limit.
---
To solve these limits, consider simplifying the function and analyzing the denominator and numerator for their behavior as \( x \) approaches -3. This will involve understanding whether the function approaches positive or negative infinity as it nears the point of interest.
For more in-depth learning, ensure to review the rules of limits, especially how functions behave around points where they might be undefined or infinite.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa451162e-14ac-4cfd-a2c6-1eae7f4a5f1a%2Fc2a690ca-dbdd-4023-95c3-a798e667f2c0%2F720ad5o_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Limit Evaluation Problem
**Problem Statement:**
45. Use the function \( f \) below to determine each limit:
\[ f(x) = \frac{-2x}{(x+3)^2} \]
**Limits to be evaluated:**
A. \[ \lim_{{x \to -3^+}} f(x) \]
B. \[ \lim_{{x \to -3^-}} f(x) \]
C. \[ \lim_{{x \to -3}} f(x) \]
---
**Explanation:**
Given the function \( f(x) = \frac{-2x}{(x+3)^2} \), we need to find the limits of the function as \( x \) approaches -3 from the right (\( -3^+ \)), from the left (\( -3^- \)), and as \( x \) approaches -3 in general.
1. **Limit from the Right (\( x \to -3^+ \))**:
- Observe the behavior of the function as \( x \) gets closer to -3 from values greater than -3.
2. **Limit from the Left (\( x \to -3^- \))**:
- Observe the behavior of the function as \( x \) gets closer to -3 from values less than -3.
3. **Two-Sided Limit (\( x \to -3 \))**:
- Determine whether the right-hand limit and left-hand limit are equal and thus conclude the two-sided limit.
---
To solve these limits, consider simplifying the function and analyzing the denominator and numerator for their behavior as \( x \) approaches -3. This will involve understanding whether the function approaches positive or negative infinity as it nears the point of interest.
For more in-depth learning, ensure to review the rules of limits, especially how functions behave around points where they might be undefined or infinite.
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