1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Limit of an absolute sequence

  1. Apr 8, 2008 #1
    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.
    Last edited: Apr 8, 2008
  2. jcsd
  3. Apr 8, 2008 #2
    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.
    Last edited: Apr 8, 2008
  4. Apr 8, 2008 #3
    sorry i meant that lim goes to 1. (fixed the typo in original post)
  5. Apr 8, 2008 #4
    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.
    Last edited: Apr 8, 2008
  6. Apr 8, 2008 #5
    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?
  7. Apr 8, 2008 #6
    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]?
  8. Apr 8, 2008 #7
    hmm yeah i thought that part of my argument was a bit shady.
    can anyone offer some insights?
  9. Apr 9, 2008 #8


    User Avatar
    Staff Emeritus
    Science Advisor

    The real numbers are dense in what?

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Limit of an absolute sequence
  1. Absolute limits (Replies: 1)

  2. Limit of a Sequence (Replies: 18)

  3. Limits and Sequences (Replies: 8)

  4. Limit of a Sequence (Replies: 1)