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Limit in several variables

  1. May 12, 2013 #1
    Evaluate the limit or prove that it does not exist..

    f(x,y) -> (0,0)
    3xy/((x^2)+(4y^2))

    The attempt at a solution:

    Set x to 0 and you get 0
    set y to 0 and you get 0
    set y=x and you get 3x^2/5x^2 = 3/5
    This means that limit does not exist.

    Is this correct?
    If this is correct, how do you know that you have to set y=x? Is there a generic approach to these kinds of problems?
    Thank you for taking the time to read this and for your help.
     
  2. jcsd
  3. May 12, 2013 #2

    tiny-tim

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    welcome to pf!

    hi manjum423! welcome to pf! :smile:

    (try using the X2 button just above the Reply box :wink:)
    completely :smile:
    you try y = kx first (for constant k),

    if that doesn't work, try y = kxn

    if that doesn't work, assume the limit exists, and try to prove it!

    alternatively, use polar-ish coordinates, eg x = 2rcosθ, y = rsinθ, giving … ? :wink:
     
  4. May 12, 2013 #3

    SammyS

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    Hello manjum423 . Welcome to PF !

    We do like you to use the supplied homework template.
    1. The problem statement, all variables and given/known data

    2. Relevant equations

    3. The attempt at a solution

    You seem to have some difficulty writing you problem. It appears that you problem is something like:
    Evaluate the limit or prove that it does not exist..

    Lim(x,y)→(0,0) 3xy/((x2)+(4y2))

    Which can be displayed more nicely using LaTeX.

    ##\displaystyle \lim_{(x,y)\to(0,0)} \frac{3xy}{(x^2)+(4y^2)}##

    Your solution is correct.

    For your problem, the method you used works fine.

    You could make it a bit more general by approaching the origin along the arbitrary line , y = mx .

    That method doesn't always work either.

    One pretty good scheme is to change to polar coordinates. Then only one variable, r, goes to zero. The other, θ, remains arbitrary.
     
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