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The provided image displays a graph of the function \( f(x) = \frac{\sin[(x - 1)(x - 6)]}{(x - 1)(x - 6)} \). Below the graph, there are multiple statements asking for which you agree with. Let's analyze each of the points:

### Function Behavior:
1. **Points of Interest**: 
   - The function has discontinuities or undefined points at \( x = 1 \) and \( x = 6 \) because these points make the denominator zero. These points must be analyzed for continuity and limits.
   
2. **Behavior at \( x = 1 \) and \( x = 6 \)**: 
   - The function \( f(x) \) appears to approach finite values at both \( x = 1 \) and \( x = 6 \), but the function is likely not defined at these points due to the nature of the denominator. We would need to check for removable discontinuities (whether the limit exists).

3. **General Observations**:
   - \( f(x) \) might be continuous at points other than \( x = 1 \) and \( x = 6 \).
   - The function has smooth, periodic behavior, implying continuity everywhere else except possibly at these problematic points.

### Statement Evaluation:
1. **\( f(1) \) is defined**:
   - Since \( x = 1 \) makes the denominator zero, \( f(1) \) is likely **not defined**.

2. **\( \lim_{x \to 1} f(x) \) exists**:
   - The limit as \( x \to 1 \) might exist if the function approaches the same value from both sides (removable discontinuity). Based on the graph, this appears to be **true**.

3. **\( f(6) \) is defined**:
   - Similar to \( x = 1 \), \( f(6) \) is likely **not defined** because the denominator becomes zero at this point.

4. **\( \lim_{x \to 6} f(x) = f(6) \)**:
   - Since \( f(6) \) is not defined, this statement is **false**.

5. **The function is continuous at \( x = 6 \)**:
   - As discussed, the function is not continuous at \( x = 6 \) due to the likely undefined value at that point. This statement is **false**.

6. **The function is not continuous at \( x = 1 \)**:
   - This statement is **true** because the function is not continuous at \( x = 1 \), likely due to the undefined value.

### Conclusion:
- Select the following statements:
  - \( \lim_{x \to 1} f(x) \) exists.
  - The function is not continuous at \( x = 1 \).

Would you like further details on any of the points above? Here are five related questions to consider:

1. How do you determine if a limit exists at a point where the function is undefined?
2. What is the difference between a removable and non-removable discontinuity?
3. How can you algebraically find the limit of a function at a problematic point?
4. What does it mean for a function to be continuous at a point?
5. How does the graph of a function help in understanding its continuity?

**Tip**: Always remember that a function can have a limit at a point without being defined at that point. This often leads to removable discontinuities.

Analyze the behavior of the function f(x) = sin[(x - 1)(x - 6)] / [(x - 1)(x - 6)] based on its graph, focusing on its points of discontinuity and continuity.

Explore the behavior of the function f(x) = sin[(x - 1)(x - 6)] / [(x - 1)(x - 6)] at points of discontinuity. Understand the conditions for limits, continuity, and removable discontinuities. Suitable for students in Grades 11-12.

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