1. Let's clarify the problem: We want to find the limit of the floor of the function $f(x)$ as $x$ approaches some value $a$. 2. The floor function, denoted $\lfloor y \rfloor$, gives the greatest integer less than or equal to $y$. 3. To find $\lim_{x \to a} \lfloor f(x) \rfloor$, we first examine $\lim_{x \to a} f(x)$, if this limit exists. 4. If $\lim_{x \to a} f(x) = L$ and $L$ is not an integer, then $\lim_{x \to a} \lfloor f(x) \rfloor = \lfloor L \rfloor$ because $f(x)$ approaches $L$, so the floor approaches the floor of $L$. 5. If $L$ is an integer, the limit of $\lfloor f(x) \rfloor$ might not exist if $f(x)$ approaches $L$ from values both slightly less than $L$ and slightly greater than or equal to $L$, as the floor function could jump. 6. Therefore, more information or the specific function $f(x)$ and the point $a$ are needed to answer definitively. 7. In summary, to find $\lim_{x \to a} \lfloor f(x) \rfloor$, first find $L=\lim_{x \to a} f(x)$, then evaluate $\lfloor L \rfloor$ if $L \notin \mathbb{Z}$, else analyze behavior near $a$.