$$\mathop {\lim }\limits_{x \to 0} \left[ {\frac{{\sin \left( {\operatorname{sgn} \left( x \right)} \right)}}{{\left( {\operatorname{sgn} \left( x \right)} \right)}}} \right],$$     where $$\left[ . \right]$$ denotes the greatest integer function, is equal to :

Solution :
$$\eqalign{ & \mathop {\lim }\limits_{x \to 0 + } \left[ {\frac{{\sin \left( {\operatorname{sgn} \,x} \right)}}{{\operatorname{sgn} \left( x \right)}}} \right] = \mathop {\lim }\limits_{x \to 0 + } \left[ {\frac{{\sin \,1}}{1}} \right] = 0 \cr & = \mathop {\lim }\limits_{x \to {0^ - }} \left[ {\frac{{\sin \left( {\operatorname{sgn} \,x} \right)}}{{\operatorname{sgn} \left( x \right)}}} \right] = \mathop {\lim }\limits_{x \to {0^ - }} \left[ {\frac{{\sin \left( { - 1} \right)}}{{ - 1}}} \right] = \mathop {\lim }\limits_{x \to {0^ - }} \left[ {\sin \,1} \right] \cr & {\text{Hence, the given limit is 0}}{\text{. }} \cr} $$