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To solve the given limits using the graph of the function \( g(x) \), let's analyze each part:

1. **\(\lim\limits_{x \to -2} g(x)\)**
   - As \( x \) approaches \(-2\) from both the left and right sides, observe the behavior of the graph near \( x = -2 \).
   - From the graph, it seems that as \( x \) approaches \(-2\), the function approaches \( g(x) = 1 \).
   - **Answer: \(\lim\limits_{x \to -2} g(x) = 1\)**

2. **\(\lim\limits_{x \to 2} g(x)\)**
   - Here, examine how the function behaves as \( x \) approaches 2 from the left and right sides.
   - The graph shows a discontinuity at \( x = 2 \), with the left-hand limit approaching \( 0 \) and the right-hand limit approaching \( 4 \). Since the left-hand and right-hand limits are not equal, the limit does not exist.
   - **Answer: \(\lim\limits_{x \to 2} g(x)\) does not exist.**

3. **\(\lim\limits_{x \to 3} g(x)\)**
   - As \( x \) approaches \( 3 \), check how \( g(x) \) behaves.
   - The graph indicates that as \( x \) approaches \( 3 \) from either side, \( g(x) \) approaches \( 2 \).
   - **Answer: \(\lim\limits_{x \to 3} g(x) = 2\)**

4. **\(\lim\limits_{x \to 0} g(x)\)**
   - Examine the graph near \( x = 0 \).
   - The function appears to smoothly approach \( g(x) = 0 \) as \( x \) approaches \( 0 \).
   - **Answer: \(\lim\limits_{x \to 0} g(x) = 0\)**

5. **\( g(2) \)**
   - Check the value of the function at \( x = 2 \) directly from the graph.
   - There is a solid dot at \( x = 2 \) and \( g(x) = 3 \), indicating the value of the function at this point.
   - **Answer: \( g(2) = 3 \)**

Let me know if you need further details or explanations!

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**Relative Questions:**

1. What is the difference between a one-sided limit and a two-sided limit?
2. How do you determine the continuity of a function at a given point?
3. What are the conditions under which a limit does not exist?
4. How can you identify if a function has a jump discontinuity from a graph?
5. What is the importance of the epsilon-delta definition in understanding limits?

**Tip:** Always check both left-hand and right-hand limits to ensure the overall limit exists when analyzing the behavior of functions at specific points.

Use the given graph of the function g(x) to find the following limits: lim x-> -2 g(x), lim x-> 2 g(x), lim x-> 3 g(x), lim x-> 0 g(x), and g(2).

This problem explores the concept of limits using the graph of g(x). Learn how to determine limits from both sides, identify discontinuities, and calculate the function value at specific points.

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