fom
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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 UTC5, Bart Goddard wrote: >> Mohan Pawar <GoogleOnly@mpclasses.com> wrote in >> >> news:02fcfd2855fd48ff8dcfbda8e1753bbb@googlegroups.com: >> >> >> >>> On Thursday, October 3, 2013 1:01:06 PM UTC5, 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 >> >>> >> >> >> >> 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 on analysis in this link helpful:
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.

