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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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William Elliot

Posts: 1,606
Registered: 1/8/12
Re: Simple analytical properties of n/d
Posted: Mar 4, 2013 11:08 PM
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On Mon, 4 Mar 2013, Ross A. Finlayson wrote:
> 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.
>
> Thanks, that should read d e N, "d in N" or "d element of N", and
> f_d(n) = n/d.


d in N is easier read and prefered.

> 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].


The range of what?

> 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.
>

Huh? Do you mean f_d is constantly monotone increasing?
Well yes, it's linear.

> 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.


Indeed, as df_d(x)/dx = 1/d, lim(d->oo) df_d(x)/dx = 0.

> 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?
>

Yes, that's a verbose characteristic of f_d which I
you're talking about and not some arbitrary function f.



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