I need help with this homework problem please provide steps and detailson how you got the answer so I can learn and compare my work **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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Answered: 7x - 49 -if it exists. If the limit…

7x - 49 -if it exists. If the limit does not exist explain why. Justify each step by Find lim -7 x- 7 indicating the limit law(s), theorems, or definitions used.

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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