Re: [tan(sin x) - sin(tan x)]/(x^7) as x --> 0
From: Dave L. Renfro (renfr1dl_at_cmich.edu)
Date: 02/12/05
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Date: Sat, 12 Feb 2005 21:39:06 +0000 (UTC)
In the sci.math thread "[tan(sin x) - sin(tan x)]/(x^7)
as x --> 0" I've posted two references about an interesting
limit, which are repeated below. The purpose of this post
is to archive in this thread another reference that I've
come across.
Dave L. Renfro <dlrenfro@gateway.net>
[sci.math: April 16, 2000 4:09:55:000PM]
http://mathforum.org/discuss/sci.math/m/254551/254552
> Rick Mabry (Louisiana State University at Shreveport) has
> informed me that a variation of this limit appeared around
> 1993 as problem #497 (posed by Edward Aboufadel of Rutgers) in
> The College Mathematics Journal. The specific limit that
> appeared in CMJ was
>
> LIMIT(x --> 0) of [ sin(tan x) - tan(sin x) ] / ***,
>
> where *** = arcSin[ arcTan(x) ] - arcTan[ arcSin(x) ].
>
> Rick has a web page with a detailed solution to CMJ
> problem #497 at
>
> <http://www.lsus.edu/sc/math/rmabry/problems/cmj497web.htm>.
Dave L. Renfro <dlrenfro@gateway.net>
[sci.math: August 13, 2001 2:11:01:000PM]
http://mathforum.org/discuss/sci.math/m/254551/254555
> V. K. Srinivansan, "Three perspectives on the limit of a function",
> International Journal of Mathematical Education in Science and
> Technology 28 (1997), 185-196.
> [Review 1997c.01508 at <http://www.emis.de/MATH/DI/search.html>.]
>
> This paper evaluates the limit as x --> 0 of
>
> [sin(tan x) - tan(sin x)] / [arcsin(arctan x) - arctan(arcsin x)].
>
> [[ The result is 1. ]]
The quote below is from the last half of footnote 2 on p. 78 of
Vladimir I. Arnol'd, "Evolution processes and ordinary differential
equations", pp. 73-85 in Serge Tabachnikov (editor), KVANT SELECTA:
ALGEBRA AND ANALYSIS II, Mathematical World #15, American
Mathematical Society, 1999. [The Russian original appeared
in Kvant 1986 #2, pp. 13-20.]
"However, we should not underestimate the ingenuity of Newton's
predecessors. Thus Huygens and Barrow could find, say, the value
of the limit
limit as x --> 0 of
[sin(tan x) - tan(sin x)] / [arcsin(arctan x) - arctan(arcsin x)]
instantaneously from geometrical considerations (there are few
contemporary mathematicians who could evaluate this limit
within an hour)."
Dave L. Renfro
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