How to prove that f(x) = [|x - 1|/(x - 1)] * (x + 1) is discontinuous by disproving the limit as x approaches?

As the other two answers have said, discontinuity of this function follows from the nonexistence of a value for the function at x=1. Of Madhukar's 3 criteria, you appear to be wanting to prove discontinuity solely on the basis of #2.

Another way of stating this is, "There is no possible value for f(1) which would result in continuity at x=1." That can be shown by noting that

|x-1|/(x-1) =
_ { 1, x>1
_ { undefined, x=1
_ {-1, x<1

Then:
lim[x→1+] [|x-1|/(x-1)]*(x+1) = x + 1 = 2
lim[x→1-] [|x-1|/(x-1)]*(x+1) = -(x + 1) = -2

Because the directional limits at x=1 are unequal, the limit doesn't exist, and no possible value for f(1) would make f continuous there.

If you enter 1 in the x values you will see that the denominator of the first part of the equation will equal 0. By definition you cannot divide by 0 therefore it is undefined which means not existent.