To prove that lim x⟶0 [sinx/x]=0 , where [.] denotes greatest integer function . We must first prove that sin x/x tends to 1 from the values that are less than 1 as x tends to zero PROOF : Now as x tends to zero, x can take any of the values tending from right of zero and tending from left side of zero Case - 1: First we assume that x tends from right of the zero Now, If x= 0.001, sin(0.001)/0.001 = 0.999999833 If x=0.0001, sin(0.0001/0.0001) = 0.999999998 Case -2: Now we assume that x is tending from left of the zero If x= -0.001, sin(-0.00...