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

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 David Bernier Posts: 3,884 Registered: 12/13/04
Re: Sequence limit
Posted: Oct 7, 2013 11:33 PM
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On 10/07/2013 10:34 AM, Mohan Pawar scribbled:
[...]
> *********************************************************
> "Somehow" explained to Bart Goddard
> *********************************************************
> I am assuming that the most relevant issue was why limit was decided by the index m and not the base |sin(1/m)| in lim m -> 0 |sin (1/m) |^(m).
>
> The original problem was transformed into equivalent problem as below:
>
> Find lim m -> 0 |sin (1/m) |^(m)
>
> Let y = |sin (1/m) |^(m)
> Taking log on both sides
> => ln(y) = m ln(|sin (1/m)|)
> Take limit on both sides as m->0 and evaluating it
> =>lim m->0 ln(y) = lim m->0 m ln(|sin (1/m)|)= 0 (at the time of evaluating limit, m=0 is the multiplier and one doesn?t need to care about value of ln(|sin (1/m)|). Also, the limit is determinate.)
>
> =>lim m->0 ln(y) = 0
> => lim m->0 y = e^0=1
> => lim m->0 |sin (1/m) |^(m)= 1 as before. (ALSO, VERIFIABLE ON WOLFRAM ALPHA)
>
> Notice that the new additional steps are no different from my original two line justification that saves above labor except now the index m is brought down as multiplier through log operation. It is still the exponent m now as multiplier that _alone_ decided value of limit. For reference, the original two line justification from my previous post is quoted below:
>
> "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."
>

[...]

Everybody knows that your proof is leaking,
Everybody knows that it's gonna sink,
That's how it goes,
Everybody knows...

--
Let us all be paranoid. More so than no such agence, Bolon Yokte K'uh
willing.

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 GoogleOnly@mpClasses.com
10/4/13 Bart Goddard
10/4/13 GoogleOnly@mpClasses.com
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

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