Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![**Problem Statement:**
Find \(\lim_{{x \to 7}}\frac{{7x - 49}}{{|x - 7|}}\) if it exists. If the limit does not exist, explain why. **Justify each step by indicating the limit law(s), theorems, or definitions used.**
---
This problem involves evaluating a limit involving an absolute value in the denominator. To successfully address this problem, you should understand the behavior of the function as \( x \) approaches 7 from both the left and the right. Key concepts and theorems in limit theory will be pertinent to providing a full explanation.
**Step-by-step Solution:**
1. **Rewrite the Function**:
The numerator can be factored as follows:
\[
7x - 49 = 7(x - 7)
\]
Thus, the limit expression becomes:
\[
\lim_{{x \to 7}} \frac{7(x - 7)}{|x - 7|}
\]
2. **Consider Two Cases Based on the Absolute Value**:
The expression \( |x - 7| \) can be treated differently depending on whether \( x \) approaches 7 from the left (\( x \to 7^- \)) or from the right (\( x \to 7^+ \)).
- **Case 1: \( x \to 7^+ \) (approaching from the right)**:
\[
|x - 7| = x - 7
\]
The limit expression becomes:
\[
\lim_{{x \to 7^+}} \frac{7(x - 7)}{x - 7} = \lim_{{x \to 7^+}} 7 = 7
\]
Since \( x - 7 \) is positive as \( x \) approaches 7 from the right, the expression simplifies to 7.
- **Case 2: \( x \to 7^- \) (approaching from the left)**:
\[
|x - 7| = -(x - 7)
\]
The limit expression becomes:
\[
\lim_{{x \to 7^-}} \frac{7(x - 7)}{-(x - 7)} =](https://content.bartleby.com/qna-images/question/f62d5ca7-5439-4cfe-bb83-29d1381eaedb/75567282-5df5-46fb-9e6a-cee85043abc3/8ngzrcr_processed.jpeg)
Transcribed Image Text:**Problem Statement:**
Find \(\lim_{{x \to 7}}\frac{{7x - 49}}{{|x - 7|}}\) if it exists. If the limit does not exist, explain why. **Justify each step by indicating the limit law(s), theorems, or definitions used.**
---
This problem involves evaluating a limit involving an absolute value in the denominator. To successfully address this problem, you should understand the behavior of the function as \( x \) approaches 7 from both the left and the right. Key concepts and theorems in limit theory will be pertinent to providing a full explanation.
**Step-by-step Solution:**
1. **Rewrite the Function**:
The numerator can be factored as follows:
\[
7x - 49 = 7(x - 7)
\]
Thus, the limit expression becomes:
\[
\lim_{{x \to 7}} \frac{7(x - 7)}{|x - 7|}
\]
2. **Consider Two Cases Based on the Absolute Value**:
The expression \( |x - 7| \) can be treated differently depending on whether \( x \) approaches 7 from the left (\( x \to 7^- \)) or from the right (\( x \to 7^+ \)).
- **Case 1: \( x \to 7^+ \) (approaching from the right)**:
\[
|x - 7| = x - 7
\]
The limit expression becomes:
\[
\lim_{{x \to 7^+}} \frac{7(x - 7)}{x - 7} = \lim_{{x \to 7^+}} 7 = 7
\]
Since \( x - 7 \) is positive as \( x \) approaches 7 from the right, the expression simplifies to 7.
- **Case 2: \( x \to 7^- \) (approaching from the left)**:
\[
|x - 7| = -(x - 7)
\]
The limit expression becomes:
\[
\lim_{{x \to 7^-}} \frac{7(x - 7)}{-(x - 7)} =
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