Two Path Test

  1. 1. The problem statement, all variables and given/known data
    Use the two path test to prove that the following limits do not exist.


    2. Relevant equations

    [tex]\lim_{(x,y)\rightarrow{(0,0)}}\frac{4xy}{3x^2+y^2}[/tex]



    3. The attempt at a solution
    The book that I am using introduces the Two Path Test theoretically but does not show an example of how to do it, so I am a bit lost.

    Would I set x = y, and x = -y? In some of the more basic problems I was able to set x = 0 and y = 0, and find the limits would differ, proving that there was no limit. But in this case, that's obviously not possible.
     
  2. jcsd
  3. Dick

    Dick 25,735
    Science Advisor
    Homework Helper

    Those two paths look like a good choice to me. Try them out. What's the limit along each path?
     
  4. I'm getting 1 and -1, thus the limit does not exist. A question that I have that the book does not address: how do I choose the paths? Do you just try what you think will work until you find something, or is there a specific method of choosing?
     
  5. Dick

    Dick 25,735
    Science Advisor
    Homework Helper

    There's no formula for picking the paths. Just try some until you get a feeling for what's going on. Other easy ones to try are x=0 and y=0.
     
Know someone interested in this topic? Share a link to this question via email, Google+, Twitter, or Facebook

Have something to add?