Results 1 to 3 of 3

Math Help - using sequences to prove discontinuity

  1. #1
    Newbie
    Joined
    Mar 2010
    Posts
    1

    using sequences to prove discontinuity

    Question: Prove at x=0 that cos(1/x) does not have a removable discontinuity.

    Attempt: I used two sequences that as limit n-> infinity, each sequence goes to zero, but if I insert the sequences into cos(1/x), cos(1/{xn}) goes to -1 and cos(1/{yn}) goes to 1. Does this show that at x=0, cos(1/x) does not have a removable discontinuity?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,363
    Thanks
    1311
    Yes, it does. In order to have a removable discontinuity, the limit itself must exist there. If the limit exists, then the limit along any sequence converging to that point must be equal to that limit. If two different sequences, both converging to 0, give two different results, then the limit cannot exist.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by eccc2004 View Post
    Question: Prove at x=0 that cos(1/x) does not have a removable discontinuity.

    Attempt: I used two sequences that as limit n-> infinity, each sequence goes to zero, but if I insert the sequences into cos(1/x), cos(1/{xn}) goes to -1 and cos(1/{yn}) goes to 1. Does this show that at x=0, cos(1/x) does not have a removable discontinuity?

    Of course it does! it shows that \lim_{x\to \infty}=cos\left(\frac{1}{x}\right) doesn't exist and thus the discontinuity cannot be removable.

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. [SOLVED] Prove discontinuity at
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: July 8th 2011, 09:26 AM
  2. Prove or disprove sequences
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: February 22nd 2010, 12:28 AM
  3. use sequences to prove limit of |f(x)| HELP!
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: November 14th 2009, 04:40 PM
  4. Prove that the set of increasing sequences...
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: January 11th 2009, 04:30 AM
  5. Replies: 2
    Last Post: November 27th 2008, 04:44 PM

Search Tags


/mathhelpforum @mathhelpforum