Re: Limits

"Dave L. Renfro" <renfr1dl@xxxxxxxxx> wrote:
[snip]
There are 4 arithmetic indeterminate forms: oo - oo, 0/0,
oo/oo, and 0*oo.

Of course, if we explicitly show forms involving -oo, some might say that
it makes the list look unnecessarily long. But it does help to show a
correspondence between the above forms and the exponential and logarithmic
forms.

There are 3 exponential indeterminate forms:
0^0, oo^0, and 1^oo.

And the latter is taken to include 1^-oo, and so there are actually 4
exponential forms. These correspond with the 4 multiplicative forms

0*-oo, 0*oo, oo*0 and -oo*0

resp.

Less well known are the 5 logarithmic
indeterminate forms: (log_0)(0), (log_1)(1), (log_0)(oo),
(log_oo)(0), and (log_oo)(oo).

There are 5 because there would have been 5 indeterminate forms involving
division in your first list if signs had been shown. The logarithmic forms
correspond with

-oo/-oo, 0/0, oo/-oo, -oo/oo and oo/oo

resp.

The latter are so little
known that, at present, I do not believe they can be found on
the internet. However, after this post, I imagine this might
change.

I'm not sure what your definition of "indeterminate limit form" is. But I
would normally consider

1/0 (or, more generally, c/0 where c is nonzero)

to be an indeterminate limit form. After all, if all we know is that the
numerator approaches a nonzero value and the denominator approaches 0 (but
not how), we do not know whether the quotient increases without bound,
decreases without bound, or neither. [OTOH, if we operate, say, in the
one-point compactification of R, then the limit form 1/0 is determinate,
always giving unsigned infinity.]

And of course, there are many other indeterminate limit forms involving
other functions, such as the limit form floor(n) where n is integer.

David W. Cantrell
.