Limits
shapper
I am looking for a few limits problems to give to my students.
I am looking for university level. I need complex limits. I am giving
them the L'Hopital rule:
http://en.wikipedia.org/wiki/Limit_(mathematics)#l.27H.C3.B4pital.27s_rule
Does anyone knows where can I get a few problems?
And does anyone knows a few free math eBooks for downloading?
Thanks,
Miguel
Robert Israel
You might try Strang's "Calculus", available at
<http://ocw.mit.edu/ans7870/resources/Strang/strangtext.htm>.
In particular this has some nice limits exercises at the end of section 3.8.
There are lots of math books available. You might look at
<http://www.math.gatech.edu/~cain/textbooks/onlinebooks.html>
for example.
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada
Bill
"shapper" <mdm...@gmail.com> wrote in message
news:40e67f47-f976-4391...@e23g2000prf.googlegroups.com...
> Hello,
>
> I am looking for a few limits problems to give to my students.
> I am looking for university level. I need complex limits. I am giving
> them the L'Hopital rule:
> http://en.wikipedia.org/wiki/Limit_(mathematics)#l.27H.C3.B4pital.27s_rule
>
> Does anyone knows where can I get a few problems?
Your question is vague. What kind of problems are you looking for?
Applications? Problems for engineers, computer scientists, math majors?
Undergraduate or graduate level? How many do you need? Can't you just make
some up?
> And does anyone knows a few free math eBooks for downloading?
There are some at the web site of the American Mathematical Society
(www.ams.org).
Bill
>
> Thanks,
> Miguel
>
Dave L. Renfro
> I am looking for a few limits problems to give to my students.
> I am looking for university level. I need complex limits.
> I am giving them the L'Hopital rule:
> http://en.wikipedia.org/wiki/Limit_(mathematics)#l.27H.C3.B4pital.27s_rule
>
> Does anyone knows where can I get a few problems?
I've posted a few things over the years that might be of
interest. To begin with, this post gives an overview of
several elementary methods for evaluating limits:
http://mathforum.org/kb/message.jspa?messageID=225974
A really nasty limit is one that Arthur Cayley published
a short paper on back in 1885. See the following, where
I've given the complete text of that paper. Incidentally,
the journal volume date should be 1884-85, not 1894-95.
[The particular issue in that volume which contains Cayley's
paper is for April 1885. It's reprinted in Volume 12 of
his "Collected Mathematical Papers", paper #842 on pp. 319-320,
which isn't at google-books yet but probably will be in the
near future.]
http://mathforum.org/kb/message.jspa?messageID=225976
http://groups.google.com/group/sci.math/msg/455e30c4789697a7
Here are some more posts about Cayley's example. I've come
across several additional references since these posts,
but I haven't gotten around to posting them. [It's appeared
as a problem *twice* in the journal "School Science and
Mathematics", and a slightly more difficult version appeared
as a problem in an 1898 volume of "Mathesis Recueil
Mathematique", among other places.]
[tan(sin x) - sin(tan x)]/(x^7) as x --> 0
http://mathforum.org/kb/message.jspa?messageID=225975
http://mathforum.org/kb/message.jspa?messageID=225978
What follows are some things I got searching for a
few minutes through my posts of the past few years.
L'Hopital's rule: Examples and Basic Issues
http://groups.google.com/group/alt.math.undergrad/msg/0cf252d1a185845a
http://groups.google.com/group/sci.math/msg/580f7b80059e253b
http://groups.google.com/group/sci.math/msg/b0b29ac46a83679f
http://mathforum.org/kb/thread.jspa?messageID=5968462
http://mathforum.org/kb/thread.jspa?messageID=5928321
http://mathforum.org/kb/thread.jspa?messageID=684653
Exponential max/min examples
http://groups.google.com/group/sci.math/msg/e5f3cf8dc5cd64bf
A neat 1^infinity indeterminate example
http://groups.google.com/group/sci.math/msg/d88ef24febd545de
http://mathforum.org/kb/thread.jspa?messageID=5859537
lim-inf and lim-sup version of L'Hopital's rule
http://mathforum.org/kb/message.jspa?messageID=6038222
http://mathforum.org/kb/message.jspa?messageID=6039558
L'hospitals Rule with severable variables
http://groups.google.com/group/alt.math.undergrad/msg/eb8efd19eebab8f0
Assuming your e-mail address is valid (mine isn't, but it's
easy to find in some of the posts above), you'll be getting
some handouts of mine (mid to late 1990s, although they're
dated more recently) soon that I used when I taught a few
years at one of those specialized state-supported math/science
boarding high schools. One of the handouts has a lot of examples
for students to practice with. I've since extended the number
of examples to almost double as many, but I don't have access
to my files at home right now.
However, here's an example that you have to differentiate
quite a lot before you finally get to a non-indeterminate
form. (Taylor expansions of the numerator and the denominator
will show what's going on.)
limit as x --> 0 of
tan(x) - 24*tan(x/2) divided by 4*sin(x) - 15x.
Dave L. Renfro
shapper
> Miguel wrote:
> > I am looking for a few limits problems to give to my students.
> > I am looking for university level. I need complex limits.
> > I am giving them the L'Hopital rule:
>
> > Does anyone knows where can I get a few problems?
>
> I've posted a few things over the years that might be of
> interest. To begin with, this post gives an overview of
> several elementary methods for evaluating limits:
>
> http://mathforum.org/kb/message.jspa?messageID=225974
>
> A really nasty limit is one that Arthur Cayley published
> a short paper on back in 1885. See the following, where
> I've given the complete text of that paper. Incidentally,
> the journal volume date should be 1884-85, not 1894-95.
> [The particular issue in that volume which contains Cayley's
> paper is for April 1885. It's reprinted in Volume 12 of
> his "Collected Mathematical Papers", paper #842 on pp. 319-320,
> which isn't at google-books yet but probably will be in the
> near future.]
>
>
> Here are some more posts about Cayley's example. I've come
> across several additional references since these posts,
> but I haven't gotten around to posting them. [It's appeared
> as a problem *twice* in the journal "School Science and
> Mathematics", and a slightly more difficult version appeared
> as a problem in an 1898 volume of "Mathesis Recueil
> Mathematique", among other places.]
>
>
> What follows are some things I got searching for a
> few minutes through my posts of the past few years.
>
> L'Hopital's rule: Examples and Basic Issueshttp://groups.google.com/group/alt.math.undergrad/msg/0cf252d1a185845ahttp://groups.google.com/group/sci.math/msg/580f7b80059e253bhttp://groups.google.com/group/sci.math/msg/b0b29ac46a83679fhttp://mathforum.org/kb/thread.jspa?messageID=5968462http://mathforum.org/kb/thread.jspa?messageID=5928321http://mathforum.org/kb/thread.jspa?messageID=684653
>
>
> A neat 1^infinity indeterminate examplehttp://groups.google.com/group/sci.math/msg/d88ef24febd545dehttp://mathforum.org/kb/thread.jspa?messageID=5859537
>
> lim-inf and lim-sup version of L'Hopital's rulehttp://mathforum.org/kb/message.jspa?messageID=6038222http://mathforum.org/kb/message.jspa?messageID=6039558
>
> L'hospitals Rule with severable variableshttp://groups.google.com/group/alt.math.undergrad/msg/eb8efd19eebab8f0
>
> Assuming your e-mail address is valid (mine isn't, but it's
> easy to find in some of the posts above), you'll be getting
> some handouts of mine (mid to late 1990s, although they're
> dated more recently) soon that I used when I taught a few
> years at one of those specialized state-supported math/science
> boarding high schools. One of the handouts has a lot of examples
> for students to practice with. I've since extended the number
> of examples to almost double as many, but I don't have access
> to my files at home right now.
>
> However, here's an example that you have to differentiate
> quite a lot before you finally get to a non-indeterminate
> form. (Taylor expansions of the numerator and the denominator
> will show what's going on.)
>
> limit as x --> 0 of
>
> tan(x) - 24*tan(x/2) divided by 4*sin(x) - 15x.
>
> Dave L. Renfro
Hi,
Basically I am looking for problems of limits that have
indeterminations such as:
+Inf-Inf Inf/Inf 0×Inf 0/0 0^0 1^Inf
I am looking for about 100 problems to give to my students.
I found 4 or 5 but never to complex and not many.
Thanks,
Miguel
Dave L. Renfro
> Basically I am looking for problems of limits that have
> indeterminations such as:
> +Inf-Inf Inf/Inf 0×Inf 0/0 0^0 1^Inf
>
> I am looking for about 100 problems to give to my students.
>
> I found 4 or 5 but never to complex and not many.
Besides the things I gave, you can dig up quite a few
examples in old textbooks -- especially those which
go under the name "advanced calculus" and published
roughly between 1910 and 1980 or so. Your college
library should have several. I think you'll find
them around the QA 300 to QA 330 Library of Congress
locations. You can also try googling through google's
digitized books for some really old texts. You can
either use search words and terms with a date-specific
search (i.e. published between 1800 and 1910) or choose
words and/or terms that used to be used but no longer
are, such as "vanishing fractions":
http://books.google.com/books?q=vanishing+fractions
I need to go home now, where I don't currently have
internet access, so it might be a few days before I
can reply, should one be called for.
Dave L. Renfro
quasi
wrote:
>> Miguel wrote:
>> > I am looking for a few limits problems to give to my students.
>> > I am looking for university level. I need complex limits.
>> > I am giving them the L'Hopital rule:
>
>Basically I am looking for problems of limits that have
>indeterminations such as:
>+Inf-Inf Inf/Inf 0×Inf 0/0 0^0 1^Inf
>
>I am looking for about 100 problems to give to my students.
100 problems? Why? To turn them off to Calculus?
>I found 4 or 5 but never to complex and not many.
10 or 15 might be ok, but 100 for this one small topic is ridiculous.
quasi
shapper
> On Thu, 27 Dec 2007 16:33:02 -0800 (PST), shapper <mdmo...@gmail.com>
I am not going to give them the 100 problems ... just to vary ...
But ok ... 20 hard ones would be ok ...
Anyway, I am just starting to create a few ... I couldn't find
anywhere a few really good problems.
Thanks,
Miguel
The poster formerly known as Colleyville Alan
news:49ad704f-ee86-404e...@e25g2000prg.googlegroups.com...
snip
>You can also try googling through google's
>digitized books for some really old texts. You can
>either use search words and terms with a date-specific
>search (i.e. published between 1800 and 1910) or choose
>words and/or terms that used to be used but no longer
>are, such as "vanishing fractions":
If you simply search Google Books for the topic, like "Advanced Calculus" or
"L'Hosital's Rule", you will get lots of hits. If you then choose "Full
View", it will filter out those you are unable to simply download.
Primarily, you will get public domain books or digitized versions of the
Canadian Journal of Mathematics.
Here are 688 hits on Advanced Calculus:
http://books.google.com/books?as_brr=1&q=Advance+Calculus
Here are some hits on L'Hospital's Rule
http://books.google.com/books?q=L%27Hospital%27s+Rule&as_brr=1
The poster formerly known as Colleyville Alan
message news:4774626e$0$4361$4c36...@roadrunner.com...
> "Dave L. Renfro" <renf...@cmich.edu> wrote in message
> news:49ad704f-ee86-404e...@e25g2000prg.googlegroups.com...
> snip
>>You can also try googling through google's
>>digitized books for some really old texts. You can
>>either use search words and terms with a date-specific
>>search (i.e. published between 1800 and 1910) or choose
>>words and/or terms that used to be used but no longer
>>are, such as "vanishing fractions":
>
> If you simply search Google Books for the topic, like "Advanced Calculus"
> or "L'Hosital's Rule", you will get lots of hits. If you then choose
> "Full View", it will filter out those you are unable to simply download.
> Primarily, you will get public domain books or digitized versions of the
> Canadian Journal of Mathematics.
>
> Here are 688 hits on Advanced Calculus:
> http://books.google.com/books?as_brr=1&q=Advance+Calculus
It would have helped if I included the final "d" in the word "Advanced".
1,143 hits in "Full Preview mode":
http://books.google.com/books?as_brr=1&q=Advanced+Calculus
The World Wide Wade
<4cba444a-08a7-48c2...@e25g2000prg.googlegroups.com>,
shapper <mdm...@gmail.com> wrote:
You will have trouble finding good "L'Hospital's Rule problems". The
vast majority of them are better done without the use of this magic
wand.
Passerby
wrote:
>Hello,
For your consideration:
(1 - x)^(1/x) as x -> 0
x / (x^2 + 1)^(1/2) as x -> oo
[cos(x) - cos(a)] / (x - a) as x -> a
(a^x - b^x) / x as x -> 0
log[cos(2 x)] / (pi - x)^2 as x -> pi
[tan(x) - x] / [x - sin(x)] as x -> 0
[sin(x) - x] / [x - tan(x)] as x -> 0
sec(3 x) / sec(5 x) as x -> pi/2
[1 - log(x)] / x as x -> 0
log(x - pi/2) / tan(x) as x -> pi/2
For all positive and negative values of n,
x^n / e^x as x -> oo
For all positive values of m and n,
[log(x)]^m / x^n as x -> oo
For all positive values of m and n,
x^n [log(x)]^m as x -> 0
1/(x - pi) - 1/sin(x) as x -> pi
x^x as x -> 0
x^(1/x) as x -> oo
(1 + a x)^(b/x) as x -> 0
[sin(x)]^tan(x) as x -> pi/2
Have the students study the limits toward the bottom of the page
<http://mathworld.wolfram.com/LHospitalsRule.html>
for examples where L'Hospital does not hold ... and why!
Dave L. Renfro
http://groups.google.com/group/sci.math/msg/c03ee7efa4d77048
I have two things I wish to follow-up on.
---------------------------------------------------------------
First, some references to the Cayley limit problem, in
chronological order:
Arthur Cayley, "On the value of tan(sin theta) - sin(tan theta)",
Messenger of Mathematics 14 (1884-85), 191-192. [Reprinted in
Cayley's "Collected Mathematical Works", Volume 12, paper #842,
pp. 319-320.]
J. Jonesco, "Solution to Question 1153", Mathesis Recueil
Mathematique (2) 8 (1898), 100-102.
Ernest W. Hobson, A TREATISE ON PLANE AND ADVANCED TRIGONOMETRY,
7'th edition, Dover Publications, 1928/1957, xvi + 383 pages.
[See Chapter 8, p. 134, Example 3.]
Ira D. Conley, "Solution to Problem 1803", School Science and
Mathematics 43 #4 (April 1943), 387. [The result is incorrect
and there is more than one error in the approach used.]
Hugo Brandt, "Solution to Problem 2306", School Science and
Mathematics 52 #8 (November 1952), 662-663. [No mention is
made of the fact that the same problem appeared 9 years earlier.]
Curtis Cooper and Robert E. Kennedy, "Solution to Problem 7",
Missouri Journal of Mathematical Sciences 1 #3 (Fall 1989), 42-43.
Murray S. Klamkin and Laurent W. Marcoux, "Solution to Problem 497",
College Mathematics Journal 25 #2 (March 1994), 159-161.
Vilappakkam Krishnamurthy Srinivansan, "Three perspectives on
the limit of a function", International Journal of Mathematical
Education in Science and Technology 28 (1997), 185-196.
Vladimir I. Arnol'd, "Evolution processes and ordinary differential
equations", pp. 73-85 in Serge Tabachnikov (editor), KVANT SELECTRA:
ALGEBRA AND ANALYSIS II, Mathematical World #15, American Mathematical
Society, 1999.
---------------------------------------------------------------
Second, some examples.
There are 4 arithmetic indeterminate forms: oo - oo, 0/0,
oo/oo, and 0*oo. There are 3 exponential indeterminate forms:
0^0, oo^0, and 1^oo. 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). 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.
All of the examples in a manuscript whose "examples page" I
printed out and brought with me to post are exponential
indeterminate forms. However, here's an interesting 0*oo
form I came across in another manuscript/class-handout of
mine that I saw while looking for my exponential indeterminate
form examples:
limit as x --> 1 of [exp(sin b) - exp(sin bx)] * tan(pi*x/2)
is equal to (2b/pi) * exp(sin b) * (cos b)
SOME EXPONENTIAL INDETERMINATE FORMS:
1. limit as x --> 0+ of x^x is equal to 1
2. limit as x --> 0+ of x^(x^x) is equal to 0
3. To generalilze #1 and #2, limit as x --> 0+ of
x^^n is 0 or 1, according to whether the positive
integer n is odd or even.
4. limit as x --> 0+ of [ln(1 + x)] ^ x is equal to 1
5. limit as x --> oo of (1 + x^2) ^ (1/x) is equal to 1
6. limit as x --> 0 of (1 + ax)^(1/x) is equal to exp(a)
7. limit as x --> 0 of (x + a^x)^(1/x) is equal to ae
8. limit as x --> oo of (1 + ax)^(1/x) is equal to 1 {{a > 0}}
9. limit as x --> 0 of [ax + exp(bx)] ^ (c/x) is equal to exp[c(a+b)]
10. limit as x --> oo of [(x-a)/(x-b)] ^ (cx) is equal to exp[c(b-a)]
11. limit as x --> 0 of [(a^x + b^x + c^x)/3] ^ (3/x) is equal to abc
12. limit as x --> e of (ln x) ^ [1 / (x-e)] is equal to exp(1/e)
13. limit as x --> 0 of [(tan x)/x)] ^ (1/x^2) is equal to exp(1/3)
14. limit as x --> 0 of [(sin x)/x)] ^ (1/x^2) is equal to exp(-1/6)
15. limit as x --> 0 of [(sinh x)/x] ^ (1/x^2) is equal to exp(1/6)
16. limit as x --> 0 of [(sin x)/x] ^ {ax / [bx - sin(bx)} is exp(-a/
b^3)
17. limit as x --> 0 of {(1/2)*[(cos ax) + (cos bx)]} ^ (4c/x^2)
is equal to exp[-c(a^2 + b^2)]
18. limit as x --> 0 of (cos 2x) ^ (1/x^2) is equal to exp(-2)
19. limit as x--> 0 of [1 + (sin 2x)] ^ (1/x) is equal to exp(2)
20. limit as x --> 0 of [1 + 2*(sin x)] ^ (cot x) is equal to exp(2)
21. limit as x --> 0 of (cos ax) ^ [(csc bx)^2] is exp[-(a^2)/(2b^2)]
22. limit as x --> 1 of (2 - x) ^ [tan(pi*x/2)] is equal to exp(2/pi)
23. limit as x --> 0 of [(sin x)/x + x] ^ (1/x) is equal to e
24. limit as x --> pi/2 of (csc x) ^ [(tan x)^2] is equal to sqrt(e)
---------------------------------------------------------------
Dave L. Renfro
David W. Cantrell
[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
Eric Thurschwell
> ... snip ...
>
> And does anyone knows a few free math eBooks for downloading?
>
> Thanks,
> Miguel
There's a list of free math ebooks and e-lecture notes at
http://www.geocities.com/alex_stef/mylist.html#FuncAn
but you might find more using the other suggestions.
Cheers,
ET
The World Wide Wade
<8fc52279-6af8-464f...@1g2000hsl.googlegroups.com>,
"Dave L. Renfro" <renf...@cmich.edu> wrote:
> 6. limit as x --> 0 of (1 + ax)^(1/x) is equal to exp(a)
>
> 7. limit as x --> 0 of (x + a^x)^(1/x) is equal to ae
>
> 8. limit as x --> oo of (1 + ax)^(1/x) is equal to 1 {{a > 0}}
>
> 9. limit as x --> 0 of [ax + exp(bx)] ^ (c/x) is equal to exp[c(a+b)]
>
> 11. limit as x --> 0 of [(a^x + b^x + c^x)/3] ^ (3/x) is equal to abc
> 19. limit as x--> 0 of [1 + (sin 2x)] ^ (1/x) is equal to exp(2)
> 23. limit as x --> 0 of [(sin x)/x + x] ^ (1/x) is equal to e
All of these can be handled with the following result: If f(0) = 1 and
f'(0) exists, then f(x)^(1/x) -> e^(f'(0)) as x -> 0. Proof: Taking
logs, and then using the definition of the derivative, we have
ln(f(x))/x = [ln(f(x)) - ln(f(0))]/x -> (ln(f(x)))'(0) = f'(0)/f(0) =
f'(0). Now exponentiate.
Dave L. Renfro
>> 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:
Dave L. Renfro
Michael Press
<40e67f47-f976-4391...@e23g2000prf.googl
egroups.com>,
shapper <mdm...@gmail.com> wrote:
One limit recently analyzed here:
For c > 0, lim_{x -> oo} log(x)/x^c.
--
Michael Press
Dave L. Renfro
> First, some references to the Cayley limit problem, in
> chronological order:
>
> Arthur Cayley, "On the value of tan(sin theta) - sin(tan theta)",
> Messenger of Mathematics 14 (1884-85), 191-192. [Reprinted in
> Cayley's "Collected Mathematical Works", Volume 12, paper #842,
> pp. 319-320.]
Here's a challenge that some people may want to try their
hand at:
Define the n'th iterate (sin_n)(x) of the sine function
recursively by
(sin_1)(x) = (sin x)
(sin_{n+1}(x) = sin((sin_n)(x))
Then, for each positive integer n, we have
limit as x --> 0 of [ (x^(n-1)*(sin_n)(x) - (sin x)^n] / x^(n+4)
is equal to n(n-1)/36.
You can find proof by induction in the journal Mathesis Recueil
Mathematique: Series 3, Volume 2, 1902, pp. 145-147.
Other items relating to iterating the sine function, for
those interested, are:
W. Raymond Griffin, "Theory of iterated trigonometric functions",
School Science and Mathematics 45 #4 (April 1945), 341-350.
"Solution to Problem E 1437", American Mathematical Monthly
68 #5 (May 1961), 507-508.
Dave L. Renfro
Robert Israel
> Dave L. Renfro wrote (in part):
>
> > First, some references to the Cayley limit problem, in
> > chronological order:
> >
> > Arthur Cayley, "On the value of tan(sin theta) - sin(tan theta)",
> > Messenger of Mathematics 14 (1884-85), 191-192. [Reprinted in
> > Cayley's "Collected Mathematical Works", Volume 12, paper #842,
> > pp. 319-320.]
>
> Here's a challenge that some people may want to try their
> hand at:
>
> Define the n'th iterate (sin_n)(x) of the sine function
> recursively by
>
> (sin_1)(x) = (sin x)
>
> (sin_{n+1}(x) = sin((sin_n)(x))
>
> Then, for each positive integer n, we have
>
> limit as x --> 0 of [ (x^(n-1)*(sin_n)(x) - (sin x)^n] / x^(n+4)
>
> is equal to n(n-1)/36.
In fact, if Q_n(x) = ((x^(n-1)*(sin_n)(x) - (sin x)^n) / x^(n+4),
with Maple's help I get
Q_n(x) = n(n-1)/36 - n(n-1)(35n-34) x^2/3240 + O(x^4)
Dave L. Renfro
which I came across this past weekend.
Let a > 0 and b be real numbers. Then the limit as
x --> infinity of
[ (x^a) / (x^a - 1) ] ^ cot{(b/x)^a}
is equal to exp[b^(-a)].
Several methods are given in
"Solution to Question 1621", Mathesis Recueil Mathematique
(3) 7 (1907), 252-254.
The most elementary method (in the sense of how much
"theory of limits" is utilized) seems to be the following.
Let k = x^a - 1. (Note that x --> oo iff k --> oo.)
Then the expression
[ (x^a) / (x^a - 1) ] ^ cot{(b/x)^a}
becomes
[(k+1)/k] ^ (ku), where u = cot{(b/x)^a} / (x^a - 1).
Hence, [(k+1)/k] ^ (ku) = [(1 + 1/k)^k] ^ u.
As x --> oo, this approaches exp(limit as x --> oo of u).
[This appears to make use of "limit(r^s) = (limit r)^(limit s)",
a result that used to be standard in very old texts.]
Since
u = cos{(b/x)^a} * [ (b/x)^a / sin{(b/x)^a} ]
* [1 / b^a] * [ 1 / (1 - (1/x^a))],
for x --> oo we get
u --> 1 * 1 * [1 / b^a] * [1 / (1 - 0)] = b^(-a).
Therefore, the limit we want is exp[b^(-a)].
Dave L. Renfro
The World Wide Wade
<ec02b5cd-3a3c-4d99...@s8g2000prg.googlegroups.com>,
"Dave L. Renfro" <renf...@cmich.edu> wrote:
> Here's another interesting exponential indeterminate limit,
> which I came across this past weekend.
>
> Let a > 0 and b be real numbers. Then the limit as
> x --> infinity of
>
> [ (x^a) / (x^a - 1) ] ^ cot{(b/x)^a}
>
> is equal to exp[b^(-a)].
Here's one way: With c = b^a, and u = x^a, the above is the limit of
[u/(u-1)]^(cot(c/u)) = [1 + 1/(u-1)]^(cot(c/u))
as u -> oo. Take the log of this and maneuver it into
(1/c)*(u/(u-1))*{ln[1 + 1/(u-1)]/(1/(u-1))}/{tan(c/u)/(c/u)} (1).
By the definition of the derivative, both ln(1+h)/h, tan(h)/h -> 1 as
h -> 0. The limit of (1) is thus (1/c)*(1)*{1}*/{1} = 1/c = b^(-a).
Exponeniate to get the original limit.
Dave L. Renfro
http://groups.google.com/group/sci.math/msg/ae04f8e907a61bd5
http://mathforum.org/kb/message.jspa?messageID=6049132
>> Here's a challenge that some people may want to try
>> their hand at:
>>
>> Define the n'th iterate (sin_n)(x) of the sine function
>> recursively by
>>
>> (sin_1)(x) = (sin x)
>>
>> (sin_{n+1}(x) = sin((sin_n)(x))
>>
>> Then, for each positive integer n, we have
>>
>> limit as x --> 0 of [ (x^(n-1)*(sin_n)(x) - (sin x)^n] / x^(n+4)
>>
>> is equal to n(n-1)/36.
>>
>> You can find proof by induction in the journal Mathesis
>> Recueil Mathematique: Series 3, Volume 2, 1902, pp. 145-147.
>>
>> Other items relating to iterating the sine function,
>> for those interested, are:
>>
>> W. Raymond Griffin, "Theory of iterated trigonometric
>> functions", School Science and Mathematics 45 #4
>> (April 1945), 341-350.
>>
>> "Solution to Problem E 1437", American Mathematical
>> Monthly 68 #5 (May 1961), 507-508.
Robert Israel replied (31 December 2007):
http://groups.google.com/group/sci.math/msg/25d90434a44b6bc3
http://mathforum.org/kb/message.jspa?messageID=6049317&tstart=0
> In fact, if Q_n(x) = ((x^(n-1)*(sin_n)(x) - (sin x)^n) / x^(n+4),
> with Maple's help I get
> Q_n(x) = n(n-1)/36 - n(n-1)(35n-34) x^2/3240 + O(x^4)
I just came across an earlier appearance of the iterated
sine limit I posted last year, and it's freely available
on the internet:
Oscar Xavier Schlömilch, "Uebungsaufgaben für Schüler"
[Exercises for students], Archiv der Mathematik und Physik
(1) 7 (1846), 100-102. [See middle of p. 101.]
http://books.google.com/books?id=PqoKAAAAIAAJ&pg=PA101
http://books.google.com/books?id=4xIAAAAAMAAJ&pg=RA1-PA101
For what it's worth, the 1902 Mathesis Recueil Mathématique
paper I cited above is also freely available on the internet:
http://books.google.com/books?id=pSVOAAAAMAAJ&pg=PA145
The Mathesis appearance attributes the problem to
Schlömilch, but gives no reference. It's likely the
problem also appears in one of Schlömilch's textbooks
(and this may be where the Mathesis editor(s) got it),
but I haven't looked into this.
Dave L. Renfro