Re: Limits
- From: "Dave L. Renfro" <renfr1dl@xxxxxxxxx>
- Date: Thu, 27 Dec 2007 16:14:42 -0800 (PST)
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:
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
.
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