Re: Limits

Dave L. Renfro wrote (in part):

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).

David W. Cantrell wrote (in part):

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

Thanks for your comments and, if I ever put any time
into investigating this, I'll keep them in mind. A number
of years ago (not really that long, but it was probably
between 6 and 8 years ago) I came across a comment about
these logarithmic forms. I didn't jot down the reference
because I didn't expect it to be so rarely mentioned
(I've never seen any mention of this idea since then,
in fact), but I did write the idea down on a post-it-note
that I put into a folder I have in which I toss in interesting
indeterminate limit examples when I come across them (one of
over a hundred such folders of topics that have relevance
to things that often came up in my teaching or other things
that I happen to be interested in) and I came across it
yesterday when I pulled the folder out to get some examples.
I'd pretty much forgotten about it (the logarithmic forms)
until I saw it, but I'm still pretty sure I haven't come
across the idea since I wrote the note about them.

Anyway, the main reason I'm responding is about something
else that you and others (Ioannis Galikdas, Robert Israel, etc.)
might be interested in. I recently came across the following
book:

Isaac Joachim Schwatt, "An Introduction to the Operations
with Series", The Press of the University of Pennsylvania,
1924, x + 287 pages.

A second edition was published by Chelsea Publishing
Company in 1962 (which I think was just a reprint with
corrections), and reviews of it that I know of are:
Amer. Math. Monthly 32 (1925), p. 383; Mathematics
of Computation 17 (1963), pp. 91-92; Amer. Math.
Monthly 70 (1963).

This book is a virtual gold mind of all sorts of
exotic formulas. The first chapter (about 30 pages)
consists of a large number of expressions for the
n'th derivatives of various functions, and the later
chapters become more involved. Rather than try to
describe the kinds of things in this book, look at
the various conference abstracts of Schwatt's in
the Bulletin of the American Mathematical Society:

http://tinyurl.com/242q4n

Dave L. Renfro
.

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