Limit of an absolute sequence

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.

 

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

 

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?

 

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