Tuesday, 10 March 2009
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.
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