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The question provides a graph for the function \( f(x) \) and asks to determine which statements about the function at \( x = 2 \) are true. Let's go through the options step by step:

### 1. **\( f(2) \) is defined.**
   - From the graph, at \( x = 2 \), there is an open circle (meaning the function does not take a value there), so **\( f(2) \) is not defined**. This statement is **false**.

### 2. **\( \lim_{x \to 2} f(x) \) exists.**
   - The limit of \( f(x) \) as \( x \) approaches 2 from both sides is \( 3 \) (since both the left-hand and right-hand limits approach the same value). Therefore, this statement is **true**.

### 3. **\( \lim_{x \to 2} f(x) = f(2) \).**
   - Since \( f(2) \) is not defined, this statement is **false**.

### 4. **The function is continuous at \( x = 2 \).**
   - Continuity requires that:
     1. \( f(2) \) is defined.
     2. \( \lim_{x \to 2} f(x) \) exists.
     3. \( \lim_{x \to 2} f(x) = f(2) \).
   - Since \( f(2) \) is not defined, the function is **not continuous** at \( x = 2 \). This statement is **false**.

### 5. **The function is not continuous at \( x = 2 \).**
   - As discussed, \( f(2) \) is not defined, so the function is not continuous at \( x = 2 \). This statement is **true**.

### Conclusion:
The true statements are:
- \( \lim_{x \to 2} f(x) \) exists.
- The function is not continuous at \( x = 2 \).

Would you like further details or explanations?

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### 5 Related Questions:
1. What are the necessary conditions for a function to be continuous at a point?
2. How do we determine if a limit exists from a graph?
3. What is the difference between left-hand and right-hand limits?
4. Can a function have a limit at a point where it is not defined?
5. How can we use limits to classify discontinuities?

### Tip:
When checking continuity from a graph, ensure the function is defined at the point and that the limit equals the function value.

The graph below is the function f(x). Select all statements below that you agree with regarding the function at x = 2.

Explore the properties of limits and continuity for the function f(x) based on its graph at x = 2. Learn how to determine if a function is defined, if the limit exists, and whether the function is continuous at a specific point. Suitable for Grades 11-12 students.

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