$ \lim_{x \rightarrow \pi}\left \lfloor \frac{3\sin(x^2)-1}{2\cos(x)+1} \right \rfloor$=?

Question

How to find this limit: $$ \lim_{x \rightarrow \pi}\left \lfloor \frac{3\sin(x^2)-1}{2\cos(x)+1} \right \rfloor $$ where $\left \lfloor x \right \rfloor$ is the floor-function?

Generally, how to find limit of floor-function? tnx

Answer 1

Hint. The floor function $t\to\lfloor t\rfloor$ is continuous for $t\in \mathbb{R}\setminus \mathbb{Z}$. On the other hand for $n\in\mathbb{Z}$, $$\lim_{t \rightarrow n^+}\lfloor t\rfloor=n\quad\mbox{and}\quad \lim_{t \rightarrow n^-}\lfloor t\rfloor=n-1.$$ Evaluate $\displaystyle \lim_{x \rightarrow \pi}\frac{3\sin(x^2)-1}{2\cos(x)+1}.$ Is it an integer number?