Let's analyze the graph and the given statements step-by-step.

1. $f(2)$ is defined.

   • This means that there is a defined value of $f$ at $x = 2$. In the graph, at $x = 2$, there is a hole (open circle) which indicates that the function is not defined there. Hence, $f(2)$ is not defined.

2. $\lim\limits_{x \to 2} f(x)$ exists.

   • This refers to the limit of the function as $x$ approaches 2 from both sides. From the graph, as $x$ approaches 2 from the left and right, $f(x)$ approaches the same value (looks like $1$). Hence, the limit $\lim\limits_{x \to 2} f(x)$ exists.

3. $\lim\limits_{x \to 2} f(x) = f(2)$.

   • For this to be true, both the limit as $x$ approaches 2 and $f(2)$ must exist and be equal. From the previous analysis, $f(2)$ is not defined, so this statement is false.

4. The function is continuous at $x = 2$.

   • A function is continuous at $x = 2$ if $f(2)$ is defined, $\lim\limits_{x \to 2} f(x)$ exists, and $\lim\limits_{x \to 2} f(x) = f(2)$. Since $f(2)$ is not defined, the function is not continuous at $x = 2$.

5. The function is not continuous at $x = 2$.

   • Since $f(2)$ is not defined, the function is indeed not continuous at $x = 2$.

Conclusion:

The correct statements are:

• $\lim\limits_{x \to 2} f(x)$ exists.
• The function is not continuous at $x = 2$.

Would you like more details or have any other questions? Here are some related questions you might be interested in:

1. What are the different types of discontinuities in a function?
2. How do we find the limit of a function analytically?
3. What is the definition of continuity for a function?
4. Can a function have a limit at a point where it is not defined?
5. What is the graphical interpretation of a limit?

Tip: Remember, a hole in the graph indicates that the function is not defined at that point, but it does not necessarily affect the existence of the limit.