Prove that `underset(xto2)lim [x]` does not exists, where [.] represents the greatest integer function.

Options

A. `f(x)=[x]-x,c=0`
B. `f(x)=[|x|]-[2x-1],c=3`
C. `f(x)={x}^(2)-{-x}^(2),c=0`
D. `f(x)=(tan(sgn x))/((sgn x)),c=0`

Answer

A, C, D

Explanation

(A) `lim_(xrarr3)[|x|]-[2x-1]`
R.H.L `x=3-impliesf(3^(+))=3-5=-2`
L.H.L `x=3^(-)impliesf^(3^(-))=2-4=-2`
(B) `f(0^(+))=0-0=0,f(0^(-))=-1`
(C) `f(x)=(x-[x])^(2)-(-x-[-x])^(2)`
`f(0^(+))=0-(+1)^(2)=-1`
`f(0^(-))=(0-1)-(0-0)^(2)=1`
(D) `f(0^(+))=(tan1)/1=tan1`
`f(0^(-))=(tan(-1))/(-1)=tan1`