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

 Messages: [ Previous | Next ]
 quasi Posts: 12,067 Registered: 7/15/05
Re: Sequence limit
Posted: Oct 4, 2013 4:42 PM

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)

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

hence for those values of m,

|sin(1/m)|^(1/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

hence for those values of m,

|sin(1/m)|^(1/m)

approaches 1.

It follows that the limit

lim (m --> 0) |sin(1/m)|^(1/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.

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
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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