On Saturday, October 26, 2013 8:13:31 AM UTC+1, Roland Franzius wrote:
Excellent and highly relevant post.
So the situation is this:
For real numbers anyone but the most inexperienced see immediately that the limit
lim |sin (x)|^(1/x) x>inf
doesn't exist, because there are arbitrarily large values x where |sin (x)|^(1/x) is 0, and arbitrarily large values x where |sin (x)|^(1/x) is 1,
If x is restricted to integers, then it is obvious that
lim f (x)^(1/x) x->inf
is 1 if f (x) has a lower bound greater than 1 and an upper bound. Only slightly less obvious is that if f (x) has an upper bound and f (x) > 0 for all x (which it is in our case) and we take the subsequence of x's which make f (x) closer to 0 than for any smaller x, and lim f (x)^(1/x) is 1 for that subsequence, then it is 1 for the whole sequence.
Roland's post demonstrated that it is most likely that the limit is 1.