A problem with infinitesimals

bcvdbgfs

I've been playing around with a free PDF Calculus book lately. But, I have no way to check the logic used to get to a particular answer. I've been trying to find the standard part for:

(1/ɛ)((1/sqrt(4+ɛ))-(1/2))

I've tried every way I could think of to algebraically manipulate this in order to avoid dividing by zero (taking the standard part of 1/ɛ). Just by looking at the problem, I would think it would be undefined...but the odd answers tell me otherwise.

Thanks

Davidius

I'm not sure what the standard part is exactly, Wikipedia makes it out to be something similar to taking a limit. Using limits you could apply L'Hopital's rule here to get an answer, maybe something similar for this case?

Mellor

Multiplying by 1/infinity is the same as multiplying by zero is it not?

MathsManiac

Sounds like "non-standard analysis", which I know nothing about, but I agree that this is probably a formulation in that theory of what we normally consider as the limit as epsilon goes to 0.

If that's the case, l'Hopital's rule gives the answer as -1/16.

Here's how you can do it without l'Hopital's rule:

I'll use h instead of epsilon, if you don't mind, and unfortunately, I can't LaTeX!

First, bring together the two fractions in the second bracket:

(1/h)[(2-sqrt(4+h))/(2sqrt(4+h))]

Next, note that h can be written as the following, using the difference of two squares: (sqrt(4+h)-2)(sqrt(4+h)+2).

Substituting that for the h in the 1/h term allows you to cancel one of those factors (at values infinitely close, but not equal, to h=0) to get (-1)/[2(sqrt(4+h)+2)sqrt(4+h)].

This latter expression can readily be evaluated at h=0 to give -1/16.

Is that the kind of answer you were looking for?

later12

Is it not just a case of

2 - sqrt 4 + ɛ / ɛ2 sqrt 4 + ɛ

and then conjugate the numerator and multiply