https://file.aiquickdraw.com/mathbot/file/17263725840432ox552nd.jpg

The problem provides the following limits and function values:

1. \(\lim_{x \to 0} f(x) = 1\)
2. \(\lim_{x \to 3^-} f(x) = -2\) (left-hand limit as \(x\) approaches 3)
3. \(\lim_{x \to 3^+} f(x) = 2\) (right-hand limit as \(x\) approaches 3)
4. \(f(0) = -1\)
5. \(f(3) = 1\)

### Observations:
- The limit of \(f(x)\) as \(x \to 0\) is 1, but the value of the function at \(x = 0\) is \(f(0) = -1\). This suggests a **discontinuity** at \(x = 0\) because the limit does not match the actual function value.
  
- There is a **jump discontinuity** at \(x = 3\) because the left-hand limit (\(\lim_{x \to 3^-} f(x) = -2\)) and the right-hand limit (\(\lim_{x \to 3^+} f(x) = 2\)) are not equal. Moreover, the actual value of the function at \(x = 3\) is different from both limits, \(f(3) = 1\).

### Conclusion:
- The function \(f(x)\) has a discontinuity at \(x = 0\) since \(\lim_{x \to 0} f(x) \neq f(0)\).
- There is a jump discontinuity at \(x = 3\) because the left-hand limit and right-hand limit are not equal, and \(f(3)\) does not match either limit.

Do you have any questions or want further details about any part of this? 

Here are some related questions for further exploration:
1. What is the difference between removable and non-removable discontinuities?
2. How do you determine the type of discontinuity at a specific point?
3. What is the importance of one-sided limits in defining continuity?
4. Can a function have a limit at a point without being continuous there?
5. How does the value of the function at a point affect the limit?

**Tip:** When studying discontinuities, always compare the function's value at a point with the left and right limits.

The problem provides the following limits and function values: lim(x→0) f(x) = 1, lim(x→3-) f(x) = -2, lim(x→3+) f(x) = 2, f(0) = -1, f(3) = 1.

This problem explores limits and discontinuities. We analyze the function's behavior at x = 0 and x = 3 using left-hand and right-hand limits, discovering jump discontinuities and function discontinuities. Suitable for high school students in Grades 11-12.

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