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

 Messages: [ Previous | Next ]
 Bart Goddard Posts: 1,706 Registered: 12/6/04
Re: Sequence limit
Posted: Oct 4, 2013 4:22 PM

> 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
>

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 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