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

 Messages: [ Previous | Next ]
 fom Posts: 1,968 Registered: 12/4/12
Re: Sequence limit
Posted: Oct 4, 2013 9:02 PM

On 10/4/2013 6:12 PM, Mohan Pawar wrote:
> On Friday, October 4, 2013 3:22:25 PM UTC-5, Bart Goddard wrote:
>> Mohan Pawar <GoogleOnly@mpclasses.com> wrote in
>>
>>
>>
>>

>>> 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.
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>>>>
>>
>>>> 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?
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>>>>
>>
>>>
>>
>>> 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.
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>>>
>>
>>> => as x->inf., m->0
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>>> => lim x -> inf. |sin x|^(1/x)
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>>> = 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
>>
>>>
>>
>>
>>
>> The only thing that disturbs me about this is that

>>> you're an "Instructor."
>
> I am sorry if my being instructor disturbs you. However, if error/s in _my solution_ disturb you, show me at which one of the 4 steps you found an error and I will help you.
>

I would have to think the error in question
has to do with the definition for a limit and
the application of that definition.

While you will, no doubt, be far better informed
than I could ever be, you might find the remarks

http://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_power_of_zero

Your statements seem to have used this exponential
form somewhat naively, whereas the remarks in the
link suggest that you may not do so in all cases.

The examples are not quite directly relevant since
they do not involve periodic functions. But, the
mention of extending f(x,y) = x^y continuously over
a domain including (0,0) ought to apply.

So, it has less to do with your "steps" than it
does with the unexpressed assumptions behind them.

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