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Topic: Simple analytical properties of n/d
Replies: 20   Last Post: Mar 11, 2013 11:01 PM

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ross.finlayson@gmail.com

Posts: 1,179
Registered: 2/15/09
Re: Simple analytical properties of n/d
Posted: Mar 4, 2013 10:35 PM
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On Mar 3, 7:22 pm, William Elliot <ma...@panix.com> wrote:
> On Sun, 3 Mar 2013, Ross A. Finlayson wrote:
> > Survey:  does anybody find that:
> >    lim_d->oo lim_n->d n/d = 1

>
> > It's clear that it does, for all values of d e N.
>
> > Then, as a function f = n/d from N to R[0,1], d e N, n <= d E N, is it
> > not constant monotone increasing?

>
> Is that f(d) = n/d or f(n) = n/d?
> What's deN and dEN?
>

> > If not increasing, how is lim_n->d n/d = 1?
>
> lim(x->d) x/d = 1, because since f(x) = x/d is continuous,
> lim(x->d) x/d = d/d = 1
>
> For integer variables n, lim(n->d) f(n) is meaningless.
> Give a definition for it.  Wouldn't it be the same as f(d)?
>

> > Answer:  it is.
>
> Only if n is a real variable.


Hi,

Thanks, that should read d e N, "d in N" or "d element of N", and
f_d(n) = n/d.

Then, it's clear that as d->oo from 1, the range is 0, 1/2, and 1,
then 0, 1/3, 2/3, and 1, and so on, all the fractions or ratios
(reduced) or rational numbers in [0,1].

For each d, f is constant monotone increasing, as a non-negative real-
valued step function, for equal differences in magnitude in the domain
seeing equal differences of magnitude in the range, and
correspondingly for a natural-valued function.

Here, f is increasing, for each d. As d increases, the constant
monotone difference: decreases. In the limit, delta, for the
difference, goes to zero, d for denominator, also for: differential.

In the limit, 1/d goes to zero, but the function is still increasing,
as the limit of f exists and is greater than zero. (Let n range in
the reals and apply the intermediate value theorem, f is increasing
else it wouldn't have a limit greater than zero.)

Does n/d have these simple analytical properties?

Regards,

Ross Finlayson



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