Re: -- Wrong limits do not commute

On Sep 30, 10:37 am, Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx>
wrote:
*** T. Winter wrote:
In article <ca7bb$48e21363$82a1e228$7...@xxxxxxxxxxxxxxxx> Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> writes:
> *** T. Winter wrote:
> > In article <7105d$48e1e669$82a1e228$9...@xxxxxxxxxxxxxxxx> Han de Bruijn <Han.deBru...@xxxxxxxxxxxxxx> writes:
> > > *** T. Winter wrote:
> > ...
> > > > > Therefore, due to the existence of the inner limit, there is an
> > > > > upper bound on the values of y , meaning that it cannot assume
> > > > > all values.

> > > > Eh? You lost me here. The outer limit is over a function that does
> > > > not contain x. (The inner limit removes x.)

> > > Yes, but it does not remove the _condition_ on x and y .

> > How can there be a condition on x and y if there is no x?

> From the previous step. And of course there's still a (very large) x .

What x? As I wrote: the inner limit removes x. So after the previous
step there is no longer an x.

Keep dreaming about the x that has disappeared suddenly, mysteriously.
Tip, hint: have completed infinities even infected limits ? I hope not.

You really haven't a clue what the definition of limit means.

Here it is (in a somewhat more general form than usually stated):

Let f(x1,x2,...,xn) be given and suppose that there is a function
g(x2,...,xn) such that for all e and x2,...,xn, there is an N so that
if x1 > N, then
|f(x1,...,xn) - g(x2,...,xn)| < e.
Then we write lim_x1 f(x1,...,xn) = g(x2,...,xn). In other words, we
are introducing the new notation on the left hand side together with
the defining equation just in those cases where the above condition
holds.

This means that whereever you see lim_x1 f(x1,...,xn), you are free to
substitute g(x2,...,xn) with no change in meaning. (Vice versa, too,
of course).

I don't have any clue what you mean going on about completed
infinities. Whatever you mean, it is surely irrelevant, for all that
we are using is the definition of a particular notation together with
substitution of equal terms.

One last time, for emphasis:

The function lim_x1 f(x1,...,xn) is an (n-1)-ary function *by
definition*.


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