Sin is caught in the box

Calculus Level 4

\[\large \displaystyle \lim_{x \rightarrow 0} \left( \left\lfloor \dfrac{\sin x}{x} \right\rfloor + \left\lfloor \dfrac{\sin 2x}{x} \right\rfloor + \cdots + \left\lfloor \dfrac{\sin 10x}{x} \right\rfloor \right) = \, ? \]

Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.