The value of lim_(x -> 2) ([2 - x] + [x - 2] - x) = ? (where [.] denotes greatest integer function)

1 Answer
Jul 15, 2017

-3.

Explanation:

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.!