LIMIT OF GREATEST INTEGER FUNCTIONS & EXAMPLES

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(-0.001)/-0.001}

                                          _{= sin(0.001)/0.001=0.999999833}

                _{If   x=-0.0001,     sin(-0.0001)/-0.0001}

                                                         _{= sin(0.0001)/0.0001 =}  0.999999998

 _{Therefore from the above two cases we can see   that sinx/x tends  to 1 from the values that are less  than 1 as x ends to zero}

_{Hence Proved [}lim _{x⟶0   Sinx/x ]=0           {Since [0.....]=0}}

Example:

^limx⟶3  ( [x-3] + [3 - x] - 4 )

^limx⟶3 ( [x] - 3 + 3 + [-x] -4)

           = (3 - 3 + 3 - 4 - 4)

           =  -9