Originally posted by: eLiu
Originally posted by: Dari
Originally posted by: dullard
Originally posted by: Dari
EDIT: I'd also like to know the difference between 'indeterminate' and 'undefined'.
What is the value of 0/0? (Is it really undefined or are there an infinite number of values?)
There's a special word for stuff like this, where you could conceivably give it any number of values. That word is "indeterminate." It's not the same as undefined. It essentially means that if it pops up somewhere, you don't know what its value will be in your case. For instance, if you have the limit as x->0 of x/x and of 7x/x, the expression will have a value of 1 in the first case and 7 in the second case. Indeterminate.
- Dr. Robert
Link.
0 / 0 is defined. It is just that its value depends on the problem. Thus, we can't determine the value until we know the problem. We call it "indeterminate". If we used "undefined", it would mean that there will be an answer if the world has an agreement as to what the answer should be, but until then, there is no defined agreement.
OK then can you cite different values for different problems? I'm not talking about the limit to zero, which is easy.
Consider:
lim as x->0: (a*x)/x
What does that equal? a (formally justified by L'Hopital's rule)
Ok now plug in 0... 0/0 = a. i can pick any -infinity < a < infinity and I have a true statement.
To get the infinity in there...
lim as x->0: (a*x)/x^2
This limit goes to infinity for any nonzero value of a.
There you go, examples of 0/0 = any real number and 0/0 = infinity.
P.S. I just know somebody is going to be a smartass and post that a*x/x = a and there's no division by 0. So I'll head those folk off at the pass:
lim as x->0: a*(2*sin(x)-sin(3*x))/(x-sin(x))
This is again, 0/0. Using L'Hopital some more, this evalulates to 25a. So 0/0 = 25*a = anything between -inf and +inf.
To make it go to infinity, replace the denominator by (x-sin(x))^2.
