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!

How do you know what delta to choose

  1. Sep 3, 2009 #1
    Find the delta for the given epsilon. lim (1/x) =1 epsilon=.07
    x->1

    2. Relevant equations



    3. The attempt at a solution
    I got to here .07526 >x-1> -.06542 so what one is me delta??
     
  2. jcsd
  3. Sep 3, 2009 #2
    I gotta run, but it will help whoever helps you if you say how you got there.
     
  4. Sep 3, 2009 #3

    Mark44

    Staff: Mentor

    You want a positive number for delta.
     
  5. Sep 3, 2009 #4
  6. Sep 3, 2009 #5
    Let [tex]\varepsilon > 0.[/tex] Assume the existence of a [tex]\delta > 0[/tex] such that [tex]0 < |x-1| < \delta.[/tex] Then

    [tex]\left|\frac{1}{x} - 1\right| = |x-1|\cdot\frac{1}{|x|} .[/tex]

    The |x-1| term already has a bound, so we need some bound on 1/|x|. Suppose that [tex]\delta \leq 1[/tex] so that
    [tex]|1| - |x| \leq |1-x| < \delta \Rightarrow |x| > 1 - \delta \Rightarrow \frac{1}{|x|} < \frac{1}{1-\delta}.[/tex]

    This implies that

    [tex]\left|\frac{1}{x} - 1\right| = |x-1|\cdot\frac{1}{|x|} < \frac{\delta}{1 - \delta}.[/tex]

    Requiring this last expression on the right to be [tex]\leq \epsilon,[/tex] we find that setting

    [tex]\delta = \min(1, \frac{\varepsilon}{\varepsilon + 1})\mbox{ ensures that } \left|\frac{1}{x} - 1\right| < \varepsilon.[/tex]

    Note: We chose delta to be less than or equal to 1 to make it easier to find a bound for 1/|x|, but we could have picked a number smaller than 1. Any number a > 1 would not work, since this would imply that |x-1| < a which would mean that x is in (1-a, 1+a) and since 1-a < 0 < 1 + a, division by zero could occur.

    Note: Plugging in epsilon = .07 gives the delta you were looking for.
     
    Last edited: Sep 3, 2009
  7. Sep 4, 2009 #6
    Oh, I see thank you snipez.
     
  8. Sep 4, 2009 #7
    but can you explain your steps a little more please, ive never seen it done like that before. up at the top how did X-1<delta turn into l1l-lxl<=l1-xl<=delta
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook