Homework Help: Limit of an absolute sequence

ohreally1234 · Apr 8, 2008

1. The problem statement, all variables and given/known data
Consider the sequence a_n = abs(sin(x))^(1/x)
Find the lim a_n if it exists

2. Relevant equations

None. This is for my calc 2 class.

3. The attempt at a solution

We are studying the sandwich theorem, so I thought 0 < M^(1/x) < abs(sin(x))^(1/x) < 1^(1/x).
(Because I assumed that sequences imply x = 1, 2, 3, 4 ..., so sin(x) never equals 0).

Since M^(1/x) and 1 both tend to 1, I reasoned a_n must go to 1.

jhicks · Apr 8, 2008

Do you mean [tex]a_{n}=|sin(n)|^{\frac{1}{n}}[/tex]? Second, [tex]\sqrt[x]{1}[/tex] does not tend to 0 as x becomes large, nor does even [tex]\sqrt[x]{\frac{1}{2}}[/tex]. That doesn't make using 1 any less valid, you just made an incorrect assumption.

ohreally1234 · Apr 8, 2008

sorry i meant that lim goes to 1. (fixed the typo in original post)

jhicks · Apr 8, 2008

You're almost there, but I'm not convinced. Can you prove the existence of an M and that [tex]\sqrt[x]{M}[/tex] goes to 1 as x becomes large? I'm not sure how much rigor is required in your class.

ohreally1234 · Apr 8, 2008

I reasoned that M exists because the real numbers are dense.
and you can prove M^(1/x) goes to 1 using the definition of the limit.
Are there any holes in my argument?

quadraphonics · Apr 8, 2008

ohreally1234 said: ↑

(Because I assumed that sequences imply x = 1, 2, 3, 4 ..., so sin(x) never equals 0).

Well, it won't equal *exactly* 0, but doesn't the density of the reals imply that it gets arbitrarily close? What is [itex]\liminf_{n\to\infty}\left|\sin(n)\right|[/itex]?

ohreally1234 · Apr 8, 2008

hmm yeah i thought that part of my argument was a bit shady.
can anyone offer some insights?

HallsofIvy · Apr 9, 2008

ohreally1234 said: ↑

I reasoned that M exists because the real numbers are dense.

The real numbers are dense in what?

and you can prove M^(1/x) goes to 1 using the definition of the limit.
Are there any holes in my argument?