Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math

Topic: Sequence limit
Replies: 72   Last Post: Nov 26, 2013 12:07 AM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
fom

Posts: 1,968
Registered: 12/4/12
Re: Sequence limit
Posted: Oct 4, 2013 9:02 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

On 10/4/2013 6:12 PM, Mohan Pawar wrote:
> On Friday, October 4, 2013 3:22:25 PM UTC-5, Bart Goddard wrote:
>> Mohan Pawar <GoogleOnly@mpclasses.com> wrote in
>>
>> news:02fcfd28-55fd-48ff-8dcf-bda8e1753bbb@googlegroups.com:
>>
>>
>>

>>> On Thursday, October 3, 2013 1:01:06 PM UTC-5, Bart Goddard wrote:
>>
>>>> This question from a colleague:
>>
>>>>
>>
>>>>
>>
>>>>
>>
>>>> What is lim_{n -> oo} |sin n|^(1/n)
>>
>>>>
>>
>>>>
>>
>>>>
>>
>>>> where n runs through the positive integers.
>>
>>>>
>>
>>>>
>>
>>>>
>>
>>>> Calculus techniques imply the answer is 1.
>>
>>>>
>>
>>>> But the same techniques imply the answer is 1
>>
>>>>
>>
>>>> if n is changed to x, a real variable, and that
>>
>>>>
>>
>>>> is not the case, since sin x =0 infinitely often.
>>
>>>>
>>
>>>>
>>
>>>>
>>
>>>> Anyone wrestled with the subtlies of this problem?
>>
>>>>
>>
>>>
>>
>>> In fact there is no subtlety if you see why the limit is 1 in the
>>
>>> implementation of calculus techniques mentioned above for both cases
>>
>>> i.e. when
>>
>>>
>>
>>> (1) variable is n =1, 2, 3?
>>
>>>
>>
>>> (2) variable is x is any real number
>>
>>>
>>
>>> I will consider first the general case of x as real that can be used
>>
>>> to get specific case when x is n
>>
>>>
>>
>>> Solution: In given lim x -> inf. |sin x|^(1/x)
>>
>>>
>>
>>> Let
>>
>>>
>>
>>> x=1/m where m is real, -inf< x and m <inf.
>>
>>>
>>
>>> => as x->inf., m->0
>>
>>> => lim x -> inf. |sin x|^(1/x)
>>
>>> = lim m -> 0 |sin (1/m) |^(m)
>>
>>> = 1 (see below why 1)
>>
>>>
>>
>>> Note that the value of |sin (1/m)| varies from 0 to to 1 BUT exponent
>>
>>> m is guaranteed to be zero as m->0. Now if m is replaced by natural
>>
>>> number n, the situation does not change |sin (1/n)|will still be
>>
>>> within 0 to 1 and limit will evaluate to due to zero in exponent.
>>
>>>
>>
>>> Mohan Pawar
>>
>>> Online Instructor, Maths/Physics
>>
>>> MP Classes LLC
>>
>>> --------------------------------------------------
>>
>>> US Central Time: 1:53 PM 10/4/2013
>>
>>>
>>
>>
>>
>> The only thing that disturbs me about this is that

>>> you're an "Instructor."
>
> I am sorry if my being instructor disturbs you. However, if error/s in _my solution_ disturb you, show me at which one of the 4 steps you found an error and I will help you.
>


I would have to think the error in question
has to do with the definition for a limit and
the application of that definition.

While you will, no doubt, be far better informed
than I could ever be, you might find the remarks
on analysis in this link helpful:

http://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_power_of_zero

Your statements seem to have used this exponential
form somewhat naively, whereas the remarks in the
link suggest that you may not do so in all cases.

The examples are not quite directly relevant since
they do not involve periodic functions. But, the
mention of extending f(x,y) = x^y continuously over
a domain including (0,0) ought to apply.

So, it has less to do with your "steps" than it
does with the unexpressed assumptions behind them.









Date Subject Author
10/3/13
Read Sequence limit
Bart Goddard
10/3/13
Read Re: Sequence limit
Karl-Olav Nyberg
10/3/13
Read Re: Sequence limit
quasi
10/3/13
Read Re: Sequence limit
quasi
10/3/13
Read Re: Sequence limit
Karl-Olav Nyberg
10/3/13
Read Re: Sequence limit
quasi
10/4/13
Read Re: Sequence limit
Roland Franzius
10/4/13
Read Re: Sequence limit
quasi
10/5/13
Read Re: Sequence limit
Roland Franzius
10/5/13
Read Re: Sequence limit
quasi
10/26/13
Read Re: Sequence limit
Roland Franzius
10/26/13
Read Re: Sequence limit
karl
10/26/13
Read Re: Sequence limit
Roland Franzius
10/26/13
Read Re: Sequence limit
gnasher729
10/27/13
Read Re: Sequence limit
karl
10/3/13
Read Re: Sequence limit
quasi
10/4/13
Read Re: Sequence limit
Leon Aigret
10/4/13
Read Re: Sequence limit
William Elliot
10/4/13
Read Re: Sequence limit
quasi
10/4/13
Read Re: Sequence limit
William Elliot
10/4/13
Read Re: Sequence limit
quasi
10/4/13
Read Re: Sequence limit
David C. Ullrich
10/4/13
Read Re: Sequence limit
Robin Chapman
10/5/13
Read Re: Sequence limit
Bart Goddard
10/4/13
Read Re: Sequence limit
GoogleOnly@mpClasses.com
10/4/13
Read Re: Sequence limit
Bart Goddard
10/4/13
Read Re: Sequence limit
GoogleOnly@mpClasses.com
10/4/13
Read Re: Sequence limit
Peter Percival
10/5/13
Read Re: Sequence limit
Virgil
10/4/13
Read Re: Sequence limit
Bart Goddard
10/6/13
Read Re: Sequence limit
David Bernier
10/6/13
Read Re: Sequence limit
Virgil
10/6/13
Read Re: Sequence limit
Bart Goddard
10/7/13
Read Re: Sequence limit
Mohan Pawar
10/7/13
Read Re: Sequence limit
Bart Goddard
10/7/13
Read Re: Sequence limit
gnasher729
10/7/13
Read Re: Sequence limit
Richard Tobin
10/7/13
Read Re: Sequence limit
Robin Chapman
10/7/13
Read Re: Sequence limit
Michael F. Stemper
10/7/13
Read Re: Sequence limit
Michael F. Stemper
10/7/13
Read Re: Sequence limit
David Bernier
10/7/13
Read Re: Sequence limit
fom
10/8/13
Read Re: Sequence limit
Virgil
10/8/13
Read Re: Sequence limit
fom
10/8/13
Read Re: Sequence limit
Virgil
10/8/13
Read Re: Sequence limit
fom
10/4/13
Read Re: Sequence limit
fom
10/4/13
Read Re: Sequence limit
quasi
10/4/13
Read Re: Sequence limit
quasi
10/9/13
Read Re: Sequence limit
Shmuel (Seymour J.) Metz
10/10/13
Read Re: Sequence limit
Bart Goddard
11/5/13
Read Re: Sequence limit
Shmuel (Seymour J.) Metz
11/6/13
Read Re: Sequence limit
Bart Goddard
11/11/13
Read Re: Sequence limit
Shmuel (Seymour J.) Metz
11/12/13
Read Re: Sequence limit
Bart Goddard
11/15/13
Read Re: Sequence limit
Shmuel (Seymour J.) Metz
11/15/13
Read Re: Sequence limit
Bart Goddard
11/6/13
Read Re: Sequence limit
Timothy Murphy
11/8/13
Read Re: Sequence limit
Bart Goddard
11/8/13
Read Re: Sequence limit
Paul
11/8/13
Read Re: Sequence limit
Bart Goddard
11/9/13
Read Re: Sequence limit
Paul
11/9/13
Read Re: Sequence limit
quasi
11/9/13
Read Re: Sequence limit
quasi
11/9/13
Read Re: Sequence limit
quasi
11/13/13
Read Re: Sequence limit
Timothy Murphy
11/13/13
Read Re: Sequence limit
quasi
11/14/13
Read Re: Sequence limit
Timothy Murphy
11/14/13
Read Re: Sequence limit
Virgil
11/14/13
Read Re: Sequence limit
Roland Franzius
11/26/13
Read Re: Sequence limit
Shmuel (Seymour J.) Metz
11/9/13
Read Re: Sequence limit
Roland Franzius
11/9/13
Read Re: Sequence limit
Paul

Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.