https://file.aiquickdraw.com/mathbot/file/1727052933084yvv05rkz.jpg

Let's analyze the graph and solve the limits based on the function \( f(x) \).

1. **\(\lim\limits_{x \to -3} f(x)\)**:
   As \( x \to -3 \), the function approaches a value from both sides. From the left and the right, the graph approaches \( f(x) = 4 \). So,
   \[
   \lim\limits_{x \to -3} f(x) = 4
   \]

2. **\(\lim\limits_{x \to 3^+} f(x)\)** (right-hand limit as \( x \to 3 \)):
   From the right of \( x = 3 \), the graph approaches \( f(x) = 5 \), so
   \[
   \lim\limits_{x \to 3^+} f(x) = 5
   \]

3. **\(\lim\limits_{x \to 3} f(x)\)**:
   The left-hand limit of \( f(x) \) as \( x \to 3 \) approaches \( f(x) = 2 \), but the right-hand limit approaches \( f(x) = 5 \). Since the left and right limits are not equal, this limit **does not exist (DNE)**.
   \[
   \lim\limits_{x \to 3} f(x) = \text{DNE}
   \]

4. **\(\lim\limits_{x \to 2} f(x)\)**:
   As \( x \to 2 \), the graph approaches \( f(x) = 3 \) from both sides, so
   \[
   \lim\limits_{x \to 2} f(x) = 3
   \]

5. **\(\lim\limits_{x \to 2^+} f(x)\)**:
   As \( x \to 2 \) from the right, the graph again approaches \( f(x) = 3 \), so
   \[
   \lim\limits_{x \to 2^+} f(x) = 3
   \]

6. **\(\lim\limits_{x \to 4} f(x)\)**:
   As \( x \to 4 \), the graph approaches \( f(x) = 4 \) from both sides, so
   \[
   \lim\limits_{x \to 4} f(x) = 4
   \]

7. **\(\lim\limits_{x \to 4^+} f(x)\)**:
   From the right of \( x = 4 \), the graph approaches \( f(x) = 4 \), so
   \[
   \lim\limits_{x \to 4^+} f(x) = 4
   \]

### Summary of Limits:
- \(\lim\limits_{x \to -3} f(x) = 4\)
- \(\lim\limits_{x \to 3^+} f(x) = 5\)
- \(\lim\limits_{x \to 3} f(x) = \text{DNE}\)
- \(\lim\limits_{x \to 2} f(x) = 3\)
- \(\lim\limits_{x \to 2^+} f(x) = 3\)
- \(\lim\limits_{x \to 4} f(x) = 4\)
- \(\lim\limits_{x \to 4^+} f(x) = 4\)

Let me know if you'd like any clarifications!

### Related Questions:
1. What happens when a function has a jump discontinuity?
2. How are left-hand and right-hand limits different from general limits?
3. Can a limit exist if the function is undefined at that point?
4. How do we define limits at infinity?
5. What are some conditions for limits to not exist?

**Tip**: Always check the behavior of the function from both the left and right sides of a point to confirm if a limit exists.

State the value of each limit, if it exists. If it does not exist, enter 'DNE'. The graph of y = f(x) is given, and limits such as lim (x -> -3) f(x), lim (x -> 3+), and others are required.

This problem involves finding limits from a graph. Key mathematical concepts such as left-hand and right-hand limits, jump discontinuities, and limit existence are covered. Suitable for calculus students in Grades 11-12.

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