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Another two variable limit

  1. Mar 17, 2007 #1
    1. The problem statement, all variables and given/known data
    Find the limit, if it exists, or show that the limit does not exist.
    [tex]lim_{(x,y)->(0,0)}[2x^{2}y/(x^4 + y^4)][/tex]

    2. Relevant equations

    3. The attempt at a solution
    Along the y-axis and the x-axis, the limit approaches 0. Along y = mx, the limit also appaches 0. So, it appears that the limit is 0. However, the answer is that the limit "does not exist."

    Should I just keep making new equations until I find where the limit does not = 0? I even tried the Squeeze Theorem...

    0<[tex][2x^{2}y/(x^4 + y^4)][/tex]<[tex]2x^2[/tex]
    because [tex]y/(x^4 + y^4)[/tex]<1
    so as x -> 0, the whole function -> 0 right?

    Why doesn't that work to prove that the limit would be 0?
     
    Last edited: Mar 17, 2007
  2. jcsd
  3. Mar 17, 2007 #2
    Along x=y, the limit becomes

    [tex]\lim _{x \to 0} \frac {2x^3}{2x^4} = \lim _{x \to 0} \frac 1x[/tex]
     
  4. Mar 17, 2007 #3
    Thanks, I got it!
     
  5. Mar 18, 2007 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Did you read my response to your first question? No matter how many curves you try you can never prove that a limit exists that way. In fact, it is possible to show that the limit is the same for all straight lines through the origin, that would still not show that the limit is the same for curved lines.
     
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