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.independent

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 ]
Bart Goddard

Posts: 1,518
Registered: 12/6/04
Re: Sequence limit
Posted: Oct 4, 2013 4:22 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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." Note that in the real variable
case, the value of |sin x|^(1/x) is zero infinitely often.
So the that limit can't be 1.

B.


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.