https://file.aiquickdraw.com/mathbot/file/1726547541615mi77tsei.jpg

Let's analyze the function \( f(x) \) from the graph provided and address the limit-related questions when \( x = 3 \).

### Key Points from the Graph:

1. **\( \lim_{x \to 3^-} f(x) \):**
   - This represents the left-hand limit as \( x \) approaches 3 from the left. From the graph, as \( x \) approaches 3 from the left, the function value approaches \( -3 \).
   - **Answer**: \( \lim_{x \to 3^-} f(x) = -3 \).

2. **\( \lim_{x \to 3^+} f(x) \):**
   - This is the right-hand limit as \( x \) approaches 3 from the right. As \( x \to 3^+ \), the graph also shows the function approaching \( -3 \).
   - **Answer**: \( \lim_{x \to 3^+} f(x) = -3 \).

3. **\( \lim_{x \to 3} f(x) \):**
   - This is the general limit as \( x \) approaches 3. Since both the left-hand and right-hand limits are equal (both are \( -3 \)), the general limit exists and is \( -3 \).
   - **Answer**: \( \lim_{x \to 3} f(x) = -3 \).

4. **\( f(3) \):**
   - This is the value of the function at \( x = 3 \). From the graph, there is a hole at \( x = 3 \), indicating that the function is not defined at this point.
   - **Answer**: \( f(3) \) is **undefined**.

### Final Answers:
1. \( \lim_{x \to 3^-} f(x) = -3 \)
2. \( \lim_{x \to 3^+} f(x) = -3 \)
3. \( \lim_{x \to 3} f(x) = -3 \)
4. \( f(3) \) is **undefined**.

Would you like more details on limits or have any other questions?

Here are some follow-up questions:
1. What is the difference between a one-sided and two-sided limit?
2. How can you determine if a function is continuous at a given point?
3. Why is \( f(3) \) undefined in this graph?
4. What conditions are needed for the existence of \( \lim_{x \to a} f(x) \)?
5. Can you describe an example where the left and right limits are not equal?

**Tip**: When analyzing limits, always check for any discontinuities like holes, jumps, or vertical asymptotes in the graph.

Answer all of the questions below about the function f(x) graphed below when x = 3.

Analyze the limits of a function graph as x approaches 3. This problem involves understanding one-sided limits, general limits, and the concept of discontinuities in a function. Suitable for students in Grades 11-12.

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