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...
Related questions
Question
![**Problem Description:**
You are tasked with finding the limit of the following expression as \( x \) approaches 4:
\[
\lim_{{x \to 4}} \frac{|x - 4|}{x - 4}
\]
---
**Detailed Explanation:**
The problem requires evaluating the limit of \( \frac{|x - 4|}{x - 4} \) as \( x \) approaches 4.
- **Absolute value function \( |x - 4| \):** This function evaluates to \( x - 4 \) when \( x > 4 \) and \( -(x - 4) \) when \( x < 4 \).
- **Behavior Analysis:**
- As \( x \to 4^+ \) (approaching from the right, \( x > 4 \)):
- \( \frac{|x - 4|}{x - 4} = \frac{x - 4}{x - 4} = 1 \)
- As \( x \to 4^- \) (approaching from the left, \( x < 4 \)):
- \( \frac{|x - 4|}{x - 4} = \frac{-(x - 4)}{x - 4} = -1 \)
- **Conclusion:**
- Since the left-hand limit \( \lim_{{x \to 4^-}} \frac{|x - 4|}{x - 4} = -1 \) is different from the right-hand limit \( \lim_{{x \to 4^+}} \frac{|x - 4|}{x - 4} = 1 \), the two-sided limit does not exist.
Thus, the answer is:
\[
\lim_{{x \to 4}} \frac{|x - 4|}{x - 4} \text{ does not exist}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe5a78c13-da5a-4255-8b88-63fe3a182f48%2Fb39b943b-b04e-4e0c-88a5-9382ed144214%2Fl1aybas_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Description:**
You are tasked with finding the limit of the following expression as \( x \) approaches 4:
\[
\lim_{{x \to 4}} \frac{|x - 4|}{x - 4}
\]
---
**Detailed Explanation:**
The problem requires evaluating the limit of \( \frac{|x - 4|}{x - 4} \) as \( x \) approaches 4.
- **Absolute value function \( |x - 4| \):** This function evaluates to \( x - 4 \) when \( x > 4 \) and \( -(x - 4) \) when \( x < 4 \).
- **Behavior Analysis:**
- As \( x \to 4^+ \) (approaching from the right, \( x > 4 \)):
- \( \frac{|x - 4|}{x - 4} = \frac{x - 4}{x - 4} = 1 \)
- As \( x \to 4^- \) (approaching from the left, \( x < 4 \)):
- \( \frac{|x - 4|}{x - 4} = \frac{-(x - 4)}{x - 4} = -1 \)
- **Conclusion:**
- Since the left-hand limit \( \lim_{{x \to 4^-}} \frac{|x - 4|}{x - 4} = -1 \) is different from the right-hand limit \( \lim_{{x \to 4^+}} \frac{|x - 4|}{x - 4} = 1 \), the two-sided limit does not exist.
Thus, the answer is:
\[
\lim_{{x \to 4}} \frac{|x - 4|}{x - 4} \text{ does not exist}
\]
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