Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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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.
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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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