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

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quasi

Posts: 9,896
Registered: 7/15/05
Re: Sequence limit
Posted: Oct 4, 2013 4:42 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

quasi wrote:
>Mohan Pawar wrote:
>>
>>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)

>
>No, it's not equal to 1.
>
>In fact, The limit
>
> lim (m --> 0) |sin(1/m)|^(1/m)


I meant: lim (m --> 0) |sin(1/m)|^m

>does not exist.
>

>>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.

>
>No. The exponent m is only guaranteed to _approach_ 0.
>
>As m approaches 0, there are infinitely many values of m such
>that sin(1/m) = 0. For those values of m,
>
> |sin(1/m)|^(1/m) = 0


I meant: |sin(1/m)|^m = 0

>hence for those values of m,
>
> |sin(1/m)|^(1/m)


I meant: |sin(1/m)|^m

>approaches 0.
>
>On the other hand, as m approaches 0, there are infinitely
>many values of m such that sin(1/m) = 1. For those values of m,
>
> |sin(1/m)|^(1/m) = 1


I meant: |sin(1/m)|^m = 1

>hence for those values of m,
>
> |sin(1/m)|^(1/m)


I meant: |sin(1/m)|^m

>approaches 1.
>
>It follows that the limit
>
> lim (m --> 0) |sin(1/m)|^(1/m)


I meant: lim (m --> 0) |sin(1/m)|^m

>does not exist.
>
>In fact, for any real constant c between 0 and 1 inclusive,
>there exists an infinite sequence of values of m approaching
>zero such that |sin(1/m)|^(1/m) = c.


I meant: such that |sin(1/m)|^m = c.

Ugh.

By obliviously copying and pasting one typo many times, I ended
up with a lot of typos.

Sorry for the confusion.

quasi


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

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