Results 1 to 4 of 4

Math Help - Epsilon-Delta Limit Definition

  1. #1
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    652
    Thanks
    2
    Awards
    1

    Epsilon-Delta Limit Definition

    Use epsilon-delta proof to show that \lim_{x\to 4} x^2 = 16

    So far, I have:

    Let \epsilon > 0

    We know that:

    |x^2 - 16| < \epsilon

    and

    |x - 4| < \delta

    Choosing \delta:

    |x^2 - 16| < \epsilon
    |(x + 4)(x - 4)| < \epsilon
    |x + 4||x - 4| < \epsilon

    From here, I'm having trouble remembering what to do.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    o_O
    o_O is offline
    Primero Espada
    o_O's Avatar
    Joined
    Mar 2008
    From
    Canada
    Posts
    1,407
    Arbitrarily assume \delta \leq 1 which implies |x- 4 | < \delta \leq 1.

    But this implies:
    -1 <  x - 4 < 1 \ \Leftrightarrow \ 7 < x + 4 < 9 \ \Rightarrow \ |x + 4| < 9

    So we know: |x+4||x-4| < 9 |x-4| < \epsilon

    Now we can see what we should let \delta be.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member Aryth's Avatar
    Joined
    Feb 2007
    From
    USA
    Posts
    652
    Thanks
    2
    Awards
    1
    I forgot that it was reasonable to insist that x would be within 1 of 4, thanks for the help.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,417
    Thanks
    1330
    Notice that you must say that \delta is the smaller of \epsilon/9 and 1.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Epsilon delta definition of limit.
    Posted in the Calculus Forum
    Replies: 5
    Last Post: October 20th 2011, 06:20 PM
  2. Replies: 3
    Last Post: February 21st 2011, 08:11 AM
  3. Epsilon delta definition of limit
    Posted in the Calculus Forum
    Replies: 3
    Last Post: December 6th 2010, 11:39 PM
  4. Prove limit with epsilon-delta definition.
    Posted in the Calculus Forum
    Replies: 3
    Last Post: August 30th 2009, 05:19 PM
  5. Replies: 2
    Last Post: October 14th 2008, 03:11 PM

Search Tags


/mathhelpforum @mathhelpforum