The problem asks to find the right-hand limit of the function $f(x) = \text{Int}(x)$ as $x$ approaches $6$, which is denoted as $\text{Re}_6$. The function $\text{Int}(x)$ represents the greatest integer less than or equal to $x$.

1. Problem Description

The problem asks to find the right-hand limit of the function $f(x) = \text{Int}(x)$ as $x$ approaches $6$, which is denoted as $\text{Re}_6$. The function $\text{Int}(x)$ represents the greatest integer less than or equal to $x$.

2. Solution Steps

The function $\text{Int}(x)$ is also known as the floor function, denoted as $\lfloor x \rfloor$. To find the right-hand limit of $f(x) = \lfloor x \rfloor$ as $x$ approaches $6$, we need to evaluate the limit as $x$ approaches $6$ from values greater than $6$.

That is, we are looking for

$\lim_{x \to 6^+} \lfloor x \rfloor$

As $x$ approaches $6$ from the right (i.e., $x$ is slightly greater than $6$), the greatest integer less than or equal to $x$ will be $6$. For example, if $x = 6.001$, then $\lfloor x \rfloor = 6$.

Therefore, $\lim_{x \to 6^+} \lfloor x \rfloor = 6$.

3. Final Answer

6