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Limit of function with 3 variables

  1. Sep 26, 2008 #1
    1. The problem statement, all variables and given/known data

    Evaluate the following limit as (x,y,z) --> (0,0,0), if it exists:


    2. Other relevant equations

    I already know that f(x,y)=xy/x^3+y^3 doesn't exist, from a previous exercise. Hence, I suspect that this limit does not exist.

    3. The attempt at a solution

    I've taken the limit as we go along the x-axis, y=0, z=0, x-->0:

    lim (x)-->0 0/x^3 = 0.

    But, from the equation, you can tell that if you let any of x, y or z=0, you will get the limit to be equal to 0. I'm out of ideas on ways to approach this. I've tried letting y=x, but I can't cancel x or z out from the equation to make the limit not equal to 0.

    I would appreciate any hints! Thanks.
  2. jcsd
  3. Sep 26, 2008 #2


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    Science Advisor
    Homework Helper

    Ok, it approaches zero along the x axis. Try approaching 0 from a different direction. Say, set x=y=z=a and let a->0.
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