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!

Help with Epsilon Delta Proof of Multivariable Limit

  1. Jun 19, 2016 #1
    1. The problem statement, all variables and given/known data
    Hey guys. I am having a little trouble answering this question. I am teaching myself calc 3 and am a little confused here (and thus cant ask a teacher). I need to find the limit as (x,y) approaches (0,1) of f(x,y) when f(x,y)=(xy-x)/(x^2+y^2-2y+1).

    2. Relevant equations
    |f(x,y)-L|<epsilon
    0<sqrt((x-a)^2+(y-b)^2))<delta
    I made L=0, assumed epsilon>0


    3. The attempt at a solution
    Looking at the answer I see that the limit does not exist; however when I do the epsilon delta proof I cant see where I went wrong because I keep getting the result that it does :( ? So I attached a picture detailing my argument and I would love for someone to tell me where I went wrong. I chose L in the epsilon delta definition to be 0 because this is what I get when I approach (0,1) along x=0, y=1, and y=x^3+1 . I am aware that the limit does not exist because if you travel along x=y^2-1 you get a value other than zero. However my only concern is why my logic is not correct in the attached image. Thanks a lot! Also if you have tips for doing these epsilon delta proofs I would love to hear them.
     

    Attached Files:

    Last edited: Jun 19, 2016
  2. jcsd
  3. Jun 19, 2016 #2

    Charles Link

    User Avatar
    Homework Helper

    You can factor the numerator and get x(y-1) . Meanwhile your denominator factors and you get x^2+(y-1)^2. If you let ## x=\epsilon ## (it approaches zero) and let ## y-1=\Delta ## you then get a simple expression for the limit in terms of ## \epsilon ## and ## \Delta ##. If you let ## \Delta=\alpha \epsilon ## the result depends on ## \alpha ##. Thereby you don't have a single limit that it converges to. And I see your error=your denominator is greater than zero but it is not greater than 1.(top line=your inequality is incorrect.)
     
  4. Jun 19, 2016 #3
    thank you so much. I completely follow you here:). Do you mind elaborating on when I can use the technique where you choose what in your expression is delta and what is epsilon? It seems pretty powerful but I just want to make sure when and how to use it. Thanks for the help
     
  5. Jun 19, 2016 #4

    Charles Link

    User Avatar
    Homework Helper

    You have two variables, x, y that are approaching a,b respectively. Let ## x-a=\epsilon ## and ## y-b=\Delta ##. The ## \epsilon ## and the ## \Delta ## both approach zero, but there's nothing that says ## \epsilon=\Delta ##. You can let ## \Delta=\alpha \epsilon ##.The constant ## \alpha ## is quite arbitrary. If you could show your expression to give an answer that is independent of ## \alpha ##, then the limit would be what you computed by evaluating the expression with the ## \epsilon ## and ## \Delta ##. Hopefully this is helpful.
     
  6. Jun 19, 2016 #5
    very helpful! thanks
     
  7. Jun 19, 2016 #6

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    The following inequality from your picture does not hold if x2 + (y - 1)2 < 1 .
    upload_2016-6-19_18-53-34.png
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Help with Epsilon Delta Proof of Multivariable Limit
Loading...