I don't know if it is correct but I simply took a limit of the above expression when H->infinity and the result is 0.
lim H->inf [sqrt(H+1)-sqrt(H-1)] = 2lim H->inf [ 1/(sqrt(H+1)+sqrt(H-1)]=0 because the the power of the denominator is 1/2 and the power of the numerator is 0 so it is a real number.
If this is not correct pls tell me and I could try something else.
H is an infinite number, so we don't really have to look for limits. You're close but not right yet!
Ok,here is the other version:
H is infinite hyperreal number
sqrt(H+1)-sqrt(H-1) = 2/[sqrt(H+1)+sqrt(H-1)]
let's denote 1/[sqrt(H+1)+sqrt(H-1)] with e (epsilon)
the denominator of e is infinite because it is the sum of two infinite positive hyperreal numbers; (H+1,H-1 are both positive infinite and their root of any degree is positive infinite as well) so 1/positive infinity = e wich is positive and infinitesimally close to 0
So the original expression becomes 2e , a positive real number multiplied by a positive infinitesimal e->0.The result is a positive infinitesimal close to 0.
We could also take the standard part of 2e:
st(2e)=2st(e)=2*0=0 so the original expression has to be close to zero and also positive
While searching for the theory that will help me resolve the problem I stumbled upon this problem in Jerome Keisler's book and it is resolved there so I think this fact ruins my work...anyway thanks for the problem.
RIpper, your second answer is correct! Well done.
Thanks strangequark.
So why is that true, but if we were looking for H-H, it's undefined if H is hyperreal? It's a bit counter intuitive...
This is a good counterintuitive example of hyperreal numbers, although much can be understood with hardly any trouble.
So far as I know (and I know little), H-H is undefined as well as H+H, epsilon/epsilon, and a few other hyperreal expressions.
H+H is infinite because it is the sum of two positive infinite hyperreal numbers,and epsilon/epsilon is finite,you cancel out the epsilons and you get 1.Check it out in Jerome Keisler's book.
My apologies...I meant H+K and/or epsilon/delta.
Find as fast as you can (just to make this easy problem harder!) what type of number sqrt(H+1)-sqrt(H-1) is, where H is a positive infinite hyperreal number. Supply your proof. Have fun!