Fill in the blanks in each of the followings:
`lim_(xto2^(+))[x]=`_____

Answer

To solve the limit problem, we need to evaluate the limit of the greatest integer function as \( x \) approaches \( 2 \) from the right side (denoted as \( 2^+ \)).

### Step-by-Step Solution:

1. **Understanding the Greatest Integer Function**: The greatest integer function, often denoted as \( \lfloor x \rfloor \), returns the largest integer less than or equal to \( x \). For example:
- \( \lfloor 0.2 \rfloor = 0 \)
- \( \lfloor 1.5 \rfloor = 1 \)
- \( \lfloor 2.1 \rfloor = 2 \)

2. **Identifying the Limit**: We need to find \( \lim_{x \to 2^+} \lfloor x \rfloor \). This means we are looking at values of \( x \) that are slightly greater than \( 2 \).

3. **Evaluating the Function Near \( 2 \)**: As \( x \) approaches \( 2 \) from the right (for instance, \( 2.1, 2.2, 2.5, \ldots \)), the greatest integer function will yield:
- \( \lfloor 2.1 \rfloor = 2 \)
- \( \lfloor 2.2 \rfloor = 2 \)
- \( \lfloor 2.5 \rfloor = 2 \)

4. **Conclusion**: Since all values of \( \lfloor x \rfloor \) for \( x \) slightly greater than \( 2 \) are equal to \( 2 \), we can conclude that:
\[ \lim_{x \to 2^+} \lfloor x \rfloor = 2 \]

### Final Answer:
\[ \lim_{x \to 2^+} [x] = 2 \]