https://r2.erweima.ai/mathbot/image/1726087924301f38w2d4t.jpg

Let's analyze the two graphs provided in your image for limits and continuity at the specified points.

### Graph 1 (Left side, problem 3):
This graph appears to show a piecewise function where there are points of interest at \( x = 3 \).

1. **Limit at \( x = 3 \)**:
   - The function appears to approach \( y = 1 \) from both sides as \( x \to 3 \). Therefore:
   \[
   \lim_{x \to 3} f(x) = 1
   \]

2. **Continuity at \( x = 3 \)**:
   - The function has a defined value at \( f(3) = 0 \) (black dot at \( (3, 0) \)).
   - However, the limit from both sides as \( x \to 3 \) is \( 1 \), not \( 0 \).
   - Since the limit does not equal the function value at \( x = 3 \), the function is **discontinuous** at \( x = 3 \).

### Graph 2 (Right side, problem 5):
This graph involves a step function with points of interest around \( x = 2 \).

1. **Limit at \( x = 2 \)**:
   - As \( x \) approaches \( 2 \) from the left (\( x \to 2^{-} \)), the function approaches \( 3 \).
   - As \( x \) approaches \( 2 \) from the right (\( x \to 2^{+} \)), the function approaches \( -3 \).
   - Therefore, the two one-sided limits do not match, and the limit at \( x = 2 \) **does not exist**.

2. **Continuity at \( x = 2 \)**:
   - Since the limit does not exist at \( x = 2 \) and the function has different values on either side of \( x = 2 \), the function is **discontinuous** at \( x = 2 \).

### Summary:
- **Graph 3**: Limit at \( x = 3 \) exists but the function is discontinuous.
- **Graph 5**: Limit at \( x = 2 \) does not exist, and the function is discontinuous.

Would you like a deeper explanation of these limits and continuity properties?

### Related questions:
1. How do you determine if a function is continuous at a point?
2. What are the conditions for a limit to exist?
3. What does it mean for a limit to be one-sided?
4. How can piecewise functions affect continuity?
5. How would you describe a jump discontinuity?

### Tip:
When checking for continuity, always verify that the limit from both sides matches the function's value at the point.

Determine the limit (if it exists) and analyze the continuity of the function based on the two graphs shown.

This problem involves analyzing the limit and continuity of piecewise functions based on given graphs. We explore the limit at specific points and check for continuity by comparing function values and limits. Suitable for high school calculus students.

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