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date let fcx7= X-3x+2, find al lim fux) X-72 bllim fux) c) lim fox) (1) X-72

date let fcx7= X-3x+2, find al lim fux) X-72 bllim fux) c) lim fox) (1) X-72

Calculus: Early Transcendentals
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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### Problem Statement Let \( f(x) = \frac{x^2 - 3x + 2}{x + 2} \), find #### a) \(\lim_{{x \to -2^-}} f(x)\) #### b) \(\lim_{{x \to -2^+}} f(x)\) #### c) \(\lim_{{x \to -2}} f(x)\) ### Instructions In this problem, we are tasked with finding the limit of the function \( f(x) \) as \( x \) approaches -2 from the left (\( x \to -2^- \)), from the right (\( x \to -2^+ \)), and from both directions (\( x \to -2 \)). Please provide detailed steps and justifications for each part of the problem in your solution. ### Detailed Breakdown 1. **Understanding the Function** - \( f(x) = \frac{x^2 - 3x + 2}{x + 2} \) - Notice that the denominator \( x + 2 \) becomes zero when \( x = -2 \), which means the function could have a vertical asymptote or a removable discontinuity at \( x = -2 \). 2. **Limits from the Left and Right** - To understand \( \lim_{{x \to -2^-}} f(x) \) and \( \lim_{{x \to -2^+}} f(x) \), observe the behavior of \( f(x) \) as \( x \) approaches -2 from either side. Factor the numerator if possible to simplify the function. 3. **Overall Limit** - For \( \lim_{{x \to -2}} f(x) \), check if the left-hand limit and right-hand limit are equal. If they are, the overall limit exists and equals this common value; otherwise, the limit does not exist. ### Additional Resources For further understanding, you can review the concepts of limits, especially focusing on one-sided and two-sided limits, and the classification of discontinuities, which can be essential in solving this problem.
Transcribed Image Text:### Problem Statement Let \( f(x) = \frac{x^2 - 3x + 2}{x + 2} \), find #### a) \(\lim_{{x \to -2^-}} f(x)\) #### b) \(\lim_{{x \to -2^+}} f(x)\) #### c) \(\lim_{{x \to -2}} f(x)\) ### Instructions In this problem, we are tasked with finding the limit of the function \( f(x) \) as \( x \) approaches -2 from the left (\( x \to -2^- \)), from the right (\( x \to -2^+ \)), and from both directions (\( x \to -2 \)). Please provide detailed steps and justifications for each part of the problem in your solution. ### Detailed Breakdown 1. **Understanding the Function** - \( f(x) = \frac{x^2 - 3x + 2}{x + 2} \) - Notice that the denominator \( x + 2 \) becomes zero when \( x = -2 \), which means the function could have a vertical asymptote or a removable discontinuity at \( x = -2 \). 2. **Limits from the Left and Right** - To understand \( \lim_{{x \to -2^-}} f(x) \) and \( \lim_{{x \to -2^+}} f(x) \), observe the behavior of \( f(x) \) as \( x \) approaches -2 from either side. Factor the numerator if possible to simplify the function. 3. **Overall Limit** - For \( \lim_{{x \to -2}} f(x) \), check if the left-hand limit and right-hand limit are equal. If they are, the overall limit exists and equals this common value; otherwise, the limit does not exist. ### Additional Resources For further understanding, you can review the concepts of limits, especially focusing on one-sided and two-sided limits, and the classification of discontinuities, which can be essential in solving this problem.
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