Inspired by Ashish Siva

$\begin{aligned} L & = \lim_{x \to 0} \left \lfloor \frac{\color{#3D99F6}{\sin x}}{x} \right \rfloor \quad \quad \small \color{#3D99F6}{\text{By Maclaurin series}} \\ & = \lim_{x \to 0} \left \lfloor \frac{1}{x} \color{#3D99F6}{\left(x - \frac{x^3}{3!} + \frac{x^5}{5!} - ... \right)} \right \rfloor \\ & = \lim_{x \to 0} \left \lfloor \color{#3D99F6}{1 - \frac{x^2}{3!} + \frac{x^4}{5!} - ... } \right \rfloor \quad \quad \small \color{#3D99F6}{\text{As the infinite polynomial } p(x) \text{ is even.} \implies \lim_{x \to 0^-} \left \lfloor p(x) \right \rfloor = \lim_{x \to 0^+} \left \lfloor p(x) \right \rfloor = L} \\ & = \boxed{0} \quad \quad \small \color{#3D99F6}{\text{Since } p(x) < 1} \end{aligned}$