Homework Help is Here – Start Your Trial Now!
Math
Calculus
Use the graph of g to find the value of each expression. (If an answer does not exist, enter DNE. y 9 (a) lim g(x) X→ 0- DNE (b) lim g(x) (c) x → 0+ DNE lim_g(x) <→0 4 2 -2¢ X X 4 X i

Use the graph of g to find the value of each expression. (If an answer does not exist, enter DNE. y 9 (a) lim g(x) X→ 0- DNE (b) lim g(x) (c) x → 0+ DNE lim_g(x) <→0 4 2 -2¢ X X 4 X i

Calculus: Early Transcendentals
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...
See similar textbooks
icon
Related questions
Question
### Understanding Limits and Function Values: Examples and Exercises #### Example Problems and Solutions: **(d) \(\lim_{x \to 2^{-}} g(x)\)** Given value: **DNE (Does Not Exist)** Status: ❌ Incorrect --- **(e) \(\lim_{x \to 2^{+}} g(x)\)** Given value: **DNE (Does Not Exist)** Status: ❌ Incorrect --- **(f) \(\lim_{x \to 2} g(x)\)** Given value: **DNE (Does Not Exist)** Status: ✅ Correct --- **(g) \(g(2)\)** Given value: **DNE (Does Not Exist)** Status: ❌ Incorrect --- **(h) \(\lim_{x \to 4} g(x)\)** Given value: **DNE (Does Not Exist)** Status: ❌ Incorrect --- ### Explanation: #### Understanding Limits: 1. **One-Sided Limits:** - \(\lim_{x \to a^{-}} f(x)\) refers to the limit of \(f(x)\) as \(x\) approaches \(a\) from the left (less than \(a\)). - \(\lim_{x \to a^{+}} f(x)\) refers to the limit of \(f(x)\) as \(x\) approaches \(a\) from the right (greater than \(a\)). 2. **Two-Sided Limits:** - \(\lim_{x \to a} f(x)\) refers to the limit of \(f(x)\) as \(x\) approaches \(a\) from both sides. This limit exists if and only if both one-sided limits exist and are equal. 3. **Function Value:** - \(f(a)\) refers to the actual value of the function at \(x = a\). #### Common Scenarios where Limits Do Not Exist (DNE): 1. If the left-hand limit and the right-hand limit are not equal. 2. If the function approaches infinity or negative infinity as \(x\) approaches the specified point. 3. If the function exhibits oscillatory behavior (e.g., \( \sin(\frac{1}{x}) \) as \(x\) approaches 0). In these examples, the responses given for parts (d), (
Transcribed Image Text:### Understanding Limits and Function Values: Examples and Exercises #### Example Problems and Solutions: **(d) \(\lim_{x \to 2^{-}} g(x)\)** Given value: **DNE (Does Not Exist)** Status: ❌ Incorrect --- **(e) \(\lim_{x \to 2^{+}} g(x)\)** Given value: **DNE (Does Not Exist)** Status: ❌ Incorrect --- **(f) \(\lim_{x \to 2} g(x)\)** Given value: **DNE (Does Not Exist)** Status: ✅ Correct --- **(g) \(g(2)\)** Given value: **DNE (Does Not Exist)** Status: ❌ Incorrect --- **(h) \(\lim_{x \to 4} g(x)\)** Given value: **DNE (Does Not Exist)** Status: ❌ Incorrect --- ### Explanation: #### Understanding Limits: 1. **One-Sided Limits:** - \(\lim_{x \to a^{-}} f(x)\) refers to the limit of \(f(x)\) as \(x\) approaches \(a\) from the left (less than \(a\)). - \(\lim_{x \to a^{+}} f(x)\) refers to the limit of \(f(x)\) as \(x\) approaches \(a\) from the right (greater than \(a\)). 2. **Two-Sided Limits:** - \(\lim_{x \to a} f(x)\) refers to the limit of \(f(x)\) as \(x\) approaches \(a\) from both sides. This limit exists if and only if both one-sided limits exist and are equal. 3. **Function Value:** - \(f(a)\) refers to the actual value of the function at \(x = a\). #### Common Scenarios where Limits Do Not Exist (DNE): 1. If the left-hand limit and the right-hand limit are not equal. 2. If the function approaches infinity or negative infinity as \(x\) approaches the specified point. 3. If the function exhibits oscillatory behavior (e.g., \( \sin(\frac{1}{x}) \) as \(x\) approaches 0). In these examples, the responses given for parts (d), (
### Calculating Limits Using a Graph To determine the value of each limit using the graph of \( g \), observe the behavior of \( g \) as \( x \) approaches the specified value. If an answer does not exist, enter DNE. #### Graph Description: - **Axes**: The graph features a Cartesian coordinate system with the \( x \)-axis ranging from \( -2 \) to \( 6 \) and the \( y \)-axis ranging from \( -4 \) to \( 6 \). - **Curve of \( g \)**: The curve \( g \) has the following notable points: - A closed dot at \( (-2, -2) \) - An open circle at \( (0, -1) \) - A closed dot at \( (2, 1) \) - An open circle at \( (3, 1.5) \) - A closed dot at \( (4, 2) \) - An open circle at \( (5, 2.5) \) - **Behavior**: - **Near \( x = 0 \)**: As \( x \) approaches 0 from the left, the value of \( g(x) \) approaches \( -1 \). Similarly, as \( x \) approaches 0 from the right, the value of \( g(x) \) also approaches \( -1 \). #### Problems and Solutions: (a) \[ \lim_{{x \to 0^-}} g(x) \] Looking at the graph from the left of \( x = 0 \), the function \( g(x) \) approaches \( -1 \) as \( x \) approaches \( 0 \). **Answer**: \[ \lim_{{x \to 0^-}} g(x) = -1 \] (b) \[ \lim_{{x \to 0^+}} g(x) \] Looking at the graph from the right of \( x = 0 \), the function \( g(x) \) also approaches \( -1 \) as \( x \) approaches \( 0 \). **Answer**: \[ \lim_{{x \to 0^+}} g(x) = -1 \] (c) \[ \lim_{{x \to 0}} g(x) \] Since both one
Transcribed Image Text:### Calculating Limits Using a Graph To determine the value of each limit using the graph of \( g \), observe the behavior of \( g \) as \( x \) approaches the specified value. If an answer does not exist, enter DNE. #### Graph Description: - **Axes**: The graph features a Cartesian coordinate system with the \( x \)-axis ranging from \( -2 \) to \( 6 \) and the \( y \)-axis ranging from \( -4 \) to \( 6 \). - **Curve of \( g \)**: The curve \( g \) has the following notable points: - A closed dot at \( (-2, -2) \) - An open circle at \( (0, -1) \) - A closed dot at \( (2, 1) \) - An open circle at \( (3, 1.5) \) - A closed dot at \( (4, 2) \) - An open circle at \( (5, 2.5) \) - **Behavior**: - **Near \( x = 0 \)**: As \( x \) approaches 0 from the left, the value of \( g(x) \) approaches \( -1 \). Similarly, as \( x \) approaches 0 from the right, the value of \( g(x) \) also approaches \( -1 \). #### Problems and Solutions: (a) \[ \lim_{{x \to 0^-}} g(x) \] Looking at the graph from the left of \( x = 0 \), the function \( g(x) \) approaches \( -1 \) as \( x \) approaches \( 0 \). **Answer**: \[ \lim_{{x \to 0^-}} g(x) = -1 \] (b) \[ \lim_{{x \to 0^+}} g(x) \] Looking at the graph from the right of \( x = 0 \), the function \( g(x) \) also approaches \( -1 \) as \( x \) approaches \( 0 \). **Answer**: \[ \lim_{{x \to 0^+}} g(x) = -1 \] (c) \[ \lim_{{x \to 0}} g(x) \] Since both one
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Precalculus
Precalculus
Calculus
ISBN:
9780135189405
Author:
Michael Sullivan
Publisher:
PEARSON
Calculus: Early Transcendental Functions
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning
SEE MORE TEXTBOOKS