sgn(x) again

Simply a question, related to absolute value. a/a-a/a What will you think about this question? Sub. a lot of numbers and get 0? How about a=0, indeterminate form?? Let's start with the vectors. A unit vector can shows you the direction, and came from u/u, which means that it's vector divided by it's length, it only lefts the direction. For the same reason, just consider a real x as a vector v = xi+j, it's direction can only be towards +ve or negitive, so that divide v by v, and so as x/x = sgn (x) The above shows one of the defination of sgn(x). then, 0/0 is still uncalculatable. Another defination is called Dichlet Integration. sgn(x)=Int[(0->x*infinity) sinxdx/x] If x = 0, the value gives us 0. Thus the original formula becames when a isn't 0: sgn(a)-1/sgn(a) =0 When a is 0, don't think that the formula becomes 0! =>0-1/0 => still infinity We can also have a pure algebra approch: a/a-a/a=a^2-a^2/aa, so it's 0 unless a=0 They're different from x^2-1/x-1, and there's still points missing, it's an important point for absolute value.