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
![The following mathematical expression is given:
\[
\lim_{{x \to 0}} \left( 3 + \frac{{|x - 0|}}{{0 - x}} \right)
\]
This limit can be evaluated as follows:
First, simplify the expression inside the parentheses:
\[
\frac{{|x - 0|}}{{0 - x}} = \frac{{|x|}}{{-x}}
\]
Now, consider the behavior of \(|x|/(-x)\) as \(x \to 0\):
- When \(x\) approaches \(0\) from the positive side (\(x \to 0^+\)), \(|x| = x\), so the expression becomes:
\[
\frac{{x}}{{-x}} = -1
\]
- When \(x\) approaches \(0\) from the negative side (\(x \to 0^-\)), \(|x| = -x\), so the expression becomes:
\[
\frac{{-x}}{{-x}} = 1
\]
Since the limit gives different values depending on the direction from which \(x\) approaches 0, the limit does not exist. Thus, the original limit is undefined.
Therefore, the solution to the limit expression is:
\[
\dots = \text{{undefined}}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdf785451-152a-464f-9db5-033c0b71374a%2Fce6bf8ca-cfce-4e69-9c2a-4091dbd4c0f6%2F8c1vcmb_processed.png&w=3840&q=75)
Transcribed Image Text:The following mathematical expression is given:
\[
\lim_{{x \to 0}} \left( 3 + \frac{{|x - 0|}}{{0 - x}} \right)
\]
This limit can be evaluated as follows:
First, simplify the expression inside the parentheses:
\[
\frac{{|x - 0|}}{{0 - x}} = \frac{{|x|}}{{-x}}
\]
Now, consider the behavior of \(|x|/(-x)\) as \(x \to 0\):
- When \(x\) approaches \(0\) from the positive side (\(x \to 0^+\)), \(|x| = x\), so the expression becomes:
\[
\frac{{x}}{{-x}} = -1
\]
- When \(x\) approaches \(0\) from the negative side (\(x \to 0^-\)), \(|x| = -x\), so the expression becomes:
\[
\frac{{-x}}{{-x}} = 1
\]
Since the limit gives different values depending on the direction from which \(x\) approaches 0, the limit does not exist. Thus, the original limit is undefined.
Therefore, the solution to the limit expression is:
\[
\dots = \text{{undefined}}
\]
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