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?

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.