https://file.aiquickdraw.com/mathbot/file/1725398949482jpob4vnf.jpg

To solve the problem, we need to analyze the given conditions and match them with the choices provided.

### 1. **Condition Analysis**:
   - The function \( g(x) \) is increasing for \( x < 1 \) and \( x > 1 \).
   - The limit \(\lim_{x \to 1} g(x) = 5\).

### 2. **Analysis of the Given Function \( g(x) \)**:
   - \( g(x) = \frac{2x^2 + 3x - 4}{x - 1} \) for \( x \neq 1 \).
   - \( g(x) = 5 \) for \( x = 1 \).

#### **Check the limit as \( x \) approaches 1:**
   - The expression \(\frac{2x^2 + 3x - 4}{x - 1}\) suggests there might be a factor that cancels out the denominator at \( x = 1 \). Let's find this factor:

     \[
     2x^2 + 3x - 4
     \]
     - Factor the numerator:
       \[
       2x^2 + 3x - 4 = (2x - 1)(x + 4)
       \]
     - The function can be rewritten as:
       \[
       g(x) = \frac{(2x - 1)(x + 4)}{x - 1}
       \]
     - As \( x \) approaches 1, the function does not simplify easily since the denominator does not cancel out with the numerator. However, if we evaluate the limit:
       \[
       \lim_{x \to 1} g(x) = \lim_{x \to 1} \frac{2x^2 + 3x - 4}{x - 1}
       \]
     - As \( x \) approaches 1, the numerator goes to 1 and the denominator goes to 0, which results in an undefined situation.

     However, since we are given that the limit \( \lim_{x \to 1} g(x) = 5 \), this implies there is a removable discontinuity and the limit might simplify to 5 when the undefined portion is considered carefully or approached by limits, but the exact cancellation is not apparent.

### 3. **Table Analysis (II)**:
   - Looking at the table values:
     - As \( x \) approaches 1 from both sides (0.999, 1.001), \( g(x) \) approaches very large values (5000 and -5000), indicating a significant jump or break at \( x = 1 \).
     - This suggests that the function values are not continuous at \( x = 1 \), which could align with the function definition and limit condition given.

### 4. **Graph Analysis (III)**:
   - The graph shows a break at \( x = 1 \) and a continuous line elsewhere, which matches with a possible piecewise definition that jumps or has a significant change at \( x = 1 \).

### **Conclusion**:
   - Based on the condition \( g(x) \) is increasing both for \( x < 1 \) and \( x > 1 \), the function's behavior around \( x = 1 \), and the graph, we can conclude:
   - **Graph III** is consistent with the given function \( g(x) \).
   - The table (II) also suggests a discontinuity or significant change at \( x = 1 \).

### **Answer**:
   - The correct answer is **D) I and III only**.

Would you like a detailed explanation on any specific part, or do you have any other questions?

---

### Relative Questions:
1. How does factoring the numerator help in understanding the function's limit behavior?
2. What are the implications of having a removable discontinuity in a function?
3. Why is it important to check both sides of the limit when evaluating discontinuities?
4. How can graph analysis help in understanding the behavior of a function near critical points?
5. What does it mean for a function to be "increasing" and how can this be determined from a graph?

### Tip:
Always verify limit behavior by evaluating the function from both sides of the point of interest to confirm any discontinuity or removable discontinuity.

Let g be a function that is increasing for x < 1 and increasing for x > 1. If lim g(x) = 5, which of the following could represent the function g?

Explore how to determine the correct function based on given limit conditions and continuity. This problem involves analyzing piecewise functions and their behavior at critical points, ideal for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation