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.

45 110 55 None of these choices 90 Limit doesn't exist