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

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 quasi Posts: 11,917 Registered: 7/15/05
Re: Sequence limit
Posted: Oct 4, 2013 4:42 PM

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