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!

Proving the limit (delta epsilon)

  1. Oct 8, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove the follow statements directly using the formal [tex] \epsilon , \delta [/tex] definition.

    [tex] \lim_{x\rightarrow 1} \frac{x + 3}{x^2 + x + 4} = \frac{2}{3} [/tex]




    2. Relevant equations



    3. The attempt at a solution

    [tex] 0 < |x - 1| < \delta \rightarrow 0 < |\frac{x + 3}{x^2 + x + 4} - \frac{2}{3}| < \epsilon [/tex]

    Not sure what to do now.

    0 < | 3(x+3) - 2(x2 + x + 4) / 3(x2+x+4) | < e
    0 < | -2x2 + x + 1 / 3(x2+x+4) | < e
    0 < | (-2x - 1)(x - 1) / 3(x2+x+4) | < e

    Now I can control (x - 1), but how do I do this?
     
  2. jcsd
  3. Oct 9, 2009 #2
    Should I have posted this in the "Calculus and Beyond" section? :/
     
  4. Oct 9, 2009 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Now you need to get a bound on [itex]|(2x+1)/3(x^2+ x+ 4)|[/itex]

    For example, you can start by assuming that |x-1|< 1 so that 0< x< 2. Then 0< 2x< 4 and 1< 2x+1< 5. Also 0< x2< 4 so 0< x2+ x< 6, 4< x2+ x+ 4< 10 and 12< 3(x2+ x+ 4)< 30. That tells you that 1/30< 1/3(x2+ x+ 4)< 1/12 and so that 1/30< (2x+1)/3(x2+ x+ 4)< 5/12.

    Now you know that |x-1|/30< (2x-1)|x-1|/3(x2+ x+ 4)< 5|x-1|/12

    Since you want to make sure that is less than [itex]\epsilon[/itex], you want [itex]5|x-1|/12<\epsilon[/itex] so you need [itex]|x-1|< 12\epsilon/5[/itex]. To make certain that |x-1|< 1 so all of that is true, take [itex]\delta[/itex] to be the smaller of 1 and [itex]12\epsilon/5|[/itex].
     
  5. Oct 14, 2009 #4
    I've been told that I should only pick values that are less than whatever x is approaching (ie. <1 in this case), is this true?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Proving the limit (delta epsilon)
  1. Epsilon Delta Proof (Replies: 10)

  2. Epsilon- Delta Proof (Replies: 2)

Loading...