
Re: Sequence limit
Posted:
Oct 4, 2013 2:53 PM


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

