To solve this, let's analyze the graph and determine if the limit of $F(x)$ as $x$ approaches 2 exists.

1. Looking at the graph near $x = 2$, observe the behavior of the function as $x$ gets close to 2 from both the left and right sides.
2. The graph appears to approach the value $y = 3$ from both sides as $x$ approaches 2.
3. Since the left-hand limit and the right-hand limit both approach the same value, the limit exists and equals 3.

Thus, the correct choice is:

A. The limit is the real number 3.

Would you like further details or explanations on this? Here are some related questions to explore:

1. How do we formally define the limit of a function at a point?
2. What are left-hand and right-hand limits, and how do they relate to the overall limit?
3. How can you determine the limit from a graph when there are discontinuities?
4. What is the significance of the limit in understanding the behavior of a function?
5. How do limits help in understanding the continuity of a function?

Tip: When evaluating limits graphically, always check the behavior from both sides of the point to confirm if they converge to the same value.