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.!
