Let, f(x)=([2-x]+[x-2]-x).
We will find the Left Hand & Right Hand Limit of f as x to2.
As x to 2-, x < 2;" preferably, 1 < x <2."
Adding -2 to the inequality, we get, -1 lt (x-2) < 0, and,
multiplying the inequality by -1, we get, 1 gt 2-x gt 0.
:. [x-2]=-1......., and,................. [2-x]=0.
rArr lim_(x to 2-) f(x)=(0+(-1)-2)=-3.......................(star_1).
As x to 2+, x gt 2;" preferably, "2 lt x lt 3.
:. 0 lt (x-2) lt 1, and, -1 lt (2-x) lt 0.
:. [2-x]=-1, ......., and,.............. [x-2]=0.
rArr lim_(x to 2+) f(x)=(-1+0-2)=-3.........................(star_2).
From (star_1) and (star_2), we conclude that,
lim_(x to 2) f(x)=lim_(x to 2) ([2-x]+[x-2]-x)=-3.
Enjoy Maths.!