https://file.aiquickdraw.com/mathbot/file/1725998028152dv4dxb84.jpg

Based on the graph you've provided, let's analyze the statements given:

### Graph Overview
The graph shows the function \( g(x) \) plotted over the domain of \( x \) from 0 to just beyond 6. Key points on the graph where behavior changes include \( x = 4 \).

### Analysis of Statements
1. **(a) \(\lim_{x \to a} g(x)\) does not exist but \( g(a) \) is defined.**
   - This statement is marked correct for \( a = 4 \), which is consistent with the graph. \( g(4) \) is defined, but the limits from the left and right do not agree (the left limit appears to approach a value near 1, and the right limit approaches 3), hence the limit at \( x = 4 \) does not exist.

2. **(b) \(\lim_{x \to a} g(x)\) exists but \( g(a) \) is not defined.**
   - No point on the graph has a defined limit but an undefined function value; hence this statement does not apply to any \( a \).

3. **(c) \(\lim_{x \to a^-} g(x)\) and \(\lim_{x \to a^+} g(x)\) both exist but \(\lim_{x \to a} g(x)\) does not exist.**
   - For \( a = 4 \), both the left-hand and right-hand limits exist but differ; the left-hand limit is around 1, and the right-hand limit is around 3. However, this statement is marked incorrect—possibly because it also asks for the smaller and larger values, which weren't provided. You should input 1 as the smaller value and 3 as the larger value.

4. **(d) \(\lim_{x \to a^+} g(x) = g(a)\) but \(\lim_{x \to a^-} g(x) \neq g(a)\).**
   - For \( a = 4 \), \( g(4) \) appears to be defined at about 3, consistent with the right-hand limit. The left-hand limit, as noted, is near 1. Thus, this statement should be true for \( a = 4 \).

### Filling the Blanks and Corrections
- **(c)** For \( a = 4 \): Smaller value = 1, Larger value = 3.
- **(d)** For \( a = 4 \).

These insights should help you review and correct the answers on the worksheet.

Would you like further details on these explanations or help with any other questions? Here are some additional questions to consider:

1. How does the definition of a limit apply to discontinuities?
2. What is the difference between a removable and a jump discontinuity?
3. Can a function have a limit at a point where it is not defined?
4. What methods are used to find left-hand and right-hand limits analytically?
5. How would you describe the behavior of \( g(x) \) at \( x = 0 \) and \( x = 6 \)?

**Tip:** When dealing with limits graphically, always check the behavior closely around the point of interest from both the left and the right to understand the type of discontinuity or continuity present.

Determine the behavior of the function g(x) based on limits and the values of g(a) as shown in the graph.

Analyze the limits of the function g(x) at various points where g(a) is defined or undefined, using the graphical behavior. The solution involves identifying discontinuities, jump discontinuities, and assessing left-hand and right-hand limits.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation