1. Limited time only! Sign up for a free 30min personal 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!

Finding Delta Algebraically Thomas' Calculus

  1. Sep 7, 2009 #1
    1. The problem statement, all variables and given/known data

    This is Example 5 in Chapter 2.3 of the above mentioned text:

    Problem:

    Prove that the [tex]\lim_{x\rightarrow2}f(x)=4[/tex] if [tex]f(x)= x^2 \text{ for }x\ne2\text{ and }f(x)=1\text{ for }x=2[/tex]


    Solution

    Step 1 Solve the inequality [tex]|f(x)-4|<\epsilon[/tex] to find an open interval containing xo = 2 on which the inequality holds for all [itex]x\ne x_0[/itex]

    For [itex]x\ne x_0 = 2[/itex], we have f(x) = x2 and the inequality to solve is
    [tex]|x^2-4|<\epsilon[/tex]

    Thus, (Author's comments in Blue) :

    [tex]|x^2-4|<\epsilon[/tex]

    [tex]-\epsilon < x^2-4<\epsilon[/tex]

    [tex]4-\epsilon < x^2 < 4 + \epsilon[/tex]

    [tex]\sqrt{4-\epsilon} < |x| < \sqrt{4 + \epsilon}[/tex] Assumes [itex]\epsilon < 4[/itex]

    [tex]\sqrt{4-\epsilon} < x < \sqrt{4 + \epsilon}[/tex] An open interval about xo=2 that solves the inequality.

    I will stop here, though I have more questions. My initial 2 questions are these:

    1.) Maybe it's obvious, but in the last step, how did he drop the abs value sign?

    and

    2.) I fail to understand the last blue comment. How does the preceding procedure take into account that xo = 2 ?


    Thanks!
     
  2. jcsd
  3. Sep 7, 2009 #2
    I'd say that since the radicals are never negative, x can't be negative so you ignore the negative part and you can drop the absolute values. Don't know about the last thing.
     
  4. Sep 7, 2009 #3

    Avodyne

    User Avatar
    Science Advisor

    In the last line, he is specifically looking to construct an open interval of x that contains x=2. Since 2 is positive, he drops the absolute value signs. By inspection, the stated internal contains x=2.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Finding Delta Algebraically Thomas' Calculus
  1. Finding Delta (Replies: 2)

  2. Finding Delta (Replies: 0)

Loading...