Re: [tan(sin x) - sin(tan x)]/(x^7) as x --> 0
Dave L. Renfro
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 <dlre...@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 <dlre...@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
Robin Chapman
> 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.
This is a great problem. Try giving it to a calculus class
who've just learned L'Hopital's rule. :-)
I once persuaded myself that under fairly natural conditions
(f and g analytic near 0, f(x) = x + O(x^2), g(x) = x + O(x^2))
that
lim_{x->0}[f(g(x)) - g(f(x))]/x^m = 1
for the appropriate m.
--
Robin Chapman, www.maths.ex.ac.uk/~rjc/rjc.html
"Elegance is an algorithm"
Iain M. Banks, _The Algebraist_
G. A. Edgar
<r...@ivorynospamtower.freeserve.co.uk> wrote:
> Dave L. Renfro wrote:
>
> > 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.
>
> This is a great problem. Try giving it to a calculus class
> who've just learned L'Hopital's rule. :-)
>
> I once persuaded myself that under fairly natural conditions
> (f and g analytic near 0, f(x) = x + O(x^2), g(x) = x + O(x^2))
> that
> lim_{x->0}[f(g(x)) - g(f(x))]/x^m = 1
> for the appropriate m.
Try f(x) = x-x^2 and g(x) = x+x^2; Why is that example "unnatural"?
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Robert Israel
Or more generally, if f(x) = x + sum_{j=2}^infty a_j x^j and
g(x) = x + sum_{j=2}^infty b_j x^j,
f(g(x)) - g(f(x)) = (a_2 b_2 (b_2 - a_2) + a_3 b_2 - a_2 b_3) x^4 + O(x^5)
so "natural" would seem to be rather special...
Robert Israel isr...@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Robin Chapman
> In article <130220051308406429%ed...@math.ohio-state.edu.invalid>,
> G. A. Edgar <ed...@math.ohio-state.edu.invalid> wrote:
>>In article <cun5vs$8fh$4...@news6.svr.pol.co.uk>, Robin Chapman
>><r...@ivorynospamtower.freeserve.co.uk> wrote:
>
>>> I once persuaded myself that under fairly natural conditions
>>> (f and g analytic near 0, f(x) = x + O(x^2), g(x) = x + O(x^2))
>>> that
>>> lim_{x->0}[f(g(x)) - g(f(x))]/x^m = 1
>>> for the appropriate m.
>
>>Try f(x) = x-x^2 and g(x) = x+x^2; Why is that example "unnatural"?
>
> Or more generally, if f(x) = x + sum_{j=2}^infty a_j x^j and
> g(x) = x + sum_{j=2}^infty b_j x^j,
> f(g(x)) - g(f(x)) = (a_2 b_2 (b_2 - a_2) + a_3 b_2 - a_2 b_3) x^4 + O(x^5)
>
> so "natural" would seem to be rather special...
Hmmm.... I should think before putting finger to keyboard .....
I was sure that I did once come up with a way of creating/solving problems
like this quite easily, but either I was mistaken or I have forgotten
something crucial :-(