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

 Messages: [ Previous | Next ]
 GoogleOnly@mpClasses.com Posts: 24 From: USA Registered: 5/23/10
Re: Sequence limit
Posted: Oct 4, 2013 2:53 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
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US Central Time: 1:53 PM 10/4/2013

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