### 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 GraphTo 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

“Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Answered: Use the graph of g to find the value of…

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

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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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), (

### 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

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