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.