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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![**Understanding Limits: An Example**
**Problem 8: Evaluate the limit**
\[ \lim_{{x \to 0}} f(x) \]
**Graphical Analysis:**
The graph provided, which is centered around the origin (0,0) on the Cartesian plane, depicts the behavior of the function \( f(x) \) as \( x \) approaches 0 from both the left and the right sides.
**Axes Details:**
- The horizontal axis (x-axis) ranges from -4 to 4.
- The vertical axis (y-axis) ranges from -4 to 4.
- Both axes display uniform increments of 1 unit.
**Function Behavior:**
- For \( x < 0 \):
- As \( x \) approaches 0 from the left (negative x-values), the graph shows that the function \( f(x) \) approaches the value of -1. There is a specific notation, represented by an open circle at the coordinate (0, -1), indicating that the exact value of the function at \( x = 0 \) is not included but approaches -1.
- For \( x > 0 \):
- As \( x \) approaches 0 from the right (positive x-values), the function \( f(x) \) approaches the value of 2. This is also indicated with an open circle at the coordinate (0, 2), suggesting that at \( x = 0 \), the function approaches the value of 2 but does not actually take this value at that point.
**Conclusion on the Limit:**
The left-hand limit (\( \lim_{{x \to 0^-}} f(x) \)) approaches -1, and the right-hand limit (\( \lim_{{x \to 0^+}} f(x) \)) approaches 2. Since these two one-sided limits are not equal, the limit of \( f(x) \) as \( x \) approaches 0 does not exist.
Hence,
\[ \lim_{{x \to 0}} f(x) \text{ does not exist.} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F76c35efe-76b6-4dad-9a58-d1a8ec4fa095%2F15ae940c-3e2b-46ce-af48-659d1073737a%2Fb080qv.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding Limits: An Example**
**Problem 8: Evaluate the limit**
\[ \lim_{{x \to 0}} f(x) \]
**Graphical Analysis:**
The graph provided, which is centered around the origin (0,0) on the Cartesian plane, depicts the behavior of the function \( f(x) \) as \( x \) approaches 0 from both the left and the right sides.
**Axes Details:**
- The horizontal axis (x-axis) ranges from -4 to 4.
- The vertical axis (y-axis) ranges from -4 to 4.
- Both axes display uniform increments of 1 unit.
**Function Behavior:**
- For \( x < 0 \):
- As \( x \) approaches 0 from the left (negative x-values), the graph shows that the function \( f(x) \) approaches the value of -1. There is a specific notation, represented by an open circle at the coordinate (0, -1), indicating that the exact value of the function at \( x = 0 \) is not included but approaches -1.
- For \( x > 0 \):
- As \( x \) approaches 0 from the right (positive x-values), the function \( f(x) \) approaches the value of 2. This is also indicated with an open circle at the coordinate (0, 2), suggesting that at \( x = 0 \), the function approaches the value of 2 but does not actually take this value at that point.
**Conclusion on the Limit:**
The left-hand limit (\( \lim_{{x \to 0^-}} f(x) \)) approaches -1, and the right-hand limit (\( \lim_{{x \to 0^+}} f(x) \)) approaches 2. Since these two one-sided limits are not equal, the limit of \( f(x) \) as \( x \) approaches 0 does not exist.
Hence,
\[ \lim_{{x \to 0}} f(x) \text{ does not exist.} \]
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