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Directional Derivatives and Limits

  1. Mar 31, 2012 #1
    How can I use the directional derivative of a two variable function to show that the limit does not exist? For example, suppose I have a function f(x,y)=g(x)/f(y) and g(a)=f(b)=0 and the limit as x and y go to a and b is 0. How would I use the directional derivative to show that the limit at (a,b) does not exist.

    So far, I have tried to take the directional derivatives of the f(x,y) at points around the (a,b), but I feel this is inconclusive because nothing is known about the function itself, other than the fact that it is undefined at (a,b).
     
  2. jcsd
  3. Mar 31, 2012 #2
    you can choose different paths, and if the paths aren't equal, the limit does not exist.
     
  4. Mar 31, 2012 #3
    That does not resolve my issue. If the only information that I have is that g(x)/f(y) is discontinuous at (a,b), a directional derivative along different paths won't show anything due to the ambiguity of the function. Furthermore, if the limit does not exist at (a,b), then I cannot use the gradient at (a,b). At least, I don't think I can.
     
  5. Mar 31, 2012 #4

    HallsofIvy

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    What you have written really does not make much since. You say "f(x,y)=g(x)/f(y)" so that you are using f both for a function of 1 variable and a function of two variables. I think what you intend would be better written "f(x,y)= g(x)/h(y)".

    In any case, it is impossible to give an answer to your question without more information as to what f and g are like close to 0.
     
  6. Mar 31, 2012 #5
    Good point and thank you. Also, I noticed that I have made a mistake. f(x,y) should equal
    g(y)/(f(x) not g(x)/f(y). As far as providing more information, g(x)/h(y) represents any function that is undefined at a particular point. The only information that is given is that the line x=a is not in the domain, and the answer can be shown with a directional derivative, although any method is acceptable; however, I must show that the limit of f(x,y) at (a,b) does not exist.

    As far as describing what f(x) and g(y) are like close to 0, I do not know, since f(x) and g(y) merely represent one variable functions, but nothing specific.
     
  7. Mar 31, 2012 #6

    micromass

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    Try to approach is coming from a direction of the x-axis and of the y-axis. Are these two limits the same?
     
  8. Mar 31, 2012 #7
    I don't understand what you mean, but the only thing I am trying to prove is that the limit as (x,y)->(a,b) does not exist for g(y)/h(x). I am trying to do this with directional derivatives, and the line x=a is not in the domain of f(x,y)=g(y)/h(x).
     
  9. Mar 31, 2012 #8
    Could I just simply show that the directional derivative at (a-1,b) in the direction of (a+1,b) is different from the directional derivative at (a-1,b+1) in the direction of (a+1,b+1)?
     
  10. Mar 31, 2012 #9
    http://www.infoocean.info/avatar2.jpg [Broken]That does not resolve my issue.
     
    Last edited by a moderator: May 5, 2017
  11. Mar 31, 2012 #10
    What issue is that?
     
    Last edited by a moderator: May 5, 2017
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