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Application of a Limit of a multivariable functionHELP

  1. Oct 2, 2009 #1
    Application of a Limit of a multivariable function..HELP!!

    1. The problem statement, all variables and given/known data

    if f (x,y) = (xy^2)/(x+y^2).
    prove that for every real number a there is a path along which f (x,y) will approache a, as (x,y) is approaching (0,0).

    2. Relevant equations

    f (x,y) = (xy^2)/(x+y^2)
    A as a element of R
    (x,y) approches a, as (x,y) approaches (0,0)

    3. The attempt at a solution

    Don't know where to begin, any help would be great!
     
  2. jcsd
  3. Oct 2, 2009 #2
    Re: Application of a Limit of a multivariable function..HELP!!

    Suggestion: Rather than trying to get "a" to be the limit, just try taking limits along various curves. Try taking the limit along the line y=kx. Try it along the parabola y=kx^2. Neither of these actually will work, but with luck they will give you a good idea.
     
  4. Oct 2, 2009 #3

    Dick

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    Re: Application of a Limit of a multivariable function..HELP!!

    How about setting f(x,y)=a and solving for a path x(y) such that f(x(y),y)=a, no matter what the value of y as y->0?
     
    Last edited: Oct 2, 2009
  5. Oct 3, 2009 #4
    Re: Application of a Limit of a multivariable function..HELP!!

    So how does setting x(y) or y(x) help me in this equation? I'm assuming I don't have to prove the actual limit of the function do I? I'm assuming its going to be zero, just from using some polar equations, x=rcosθ and y=rsinθ , and letting the function approach r . I know that showing the limits along different paths will prove if the given limit exsists..but how do we define "all paths", and what does that say about the value a? can I make a = f(x,y) = to say z? this question isn't making sense
     
  6. Oct 3, 2009 #5

    Dick

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    Re: Application of a Limit of a multivariable function..HELP!!

    Look, the limit isn't zero. Try this. Set f(x,y)=2. Put y=(1/2), then x=(-2/7). Put y=(1/4) then x=(-2/31). Put y=(1/8) then x=(-2/127). Do you see what's happening? I'm finding (x,y) points closer and closer to (0,0) such that f(x,y)=2. Does that mean the limit is 2?
     
  7. Oct 3, 2009 #6
    Re: Application of a Limit of a multivariable function..HELP!!

    Did you try my idea yet? It does work, and pretty easily, once you "guess" the right family of paths. My suggested paths should give you enough hints if you try them.

    Edited to add: Ooops, maybe I was wrong. Better stick with Dick's idea.
     
    Last edited: Oct 3, 2009
  8. Oct 3, 2009 #7
    Re: Application of a Limit of a multivariable function..HELP!!

    y is decreasing towards zero from the positive side, and x is increasing towards zero from the negative side. for the limit to exist, we have to approach a value along all paths that lead to that value, correct?
     
  9. Oct 3, 2009 #8

    Dick

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    Re: Application of a Limit of a multivariable function..HELP!!

    You have to approach the SAME value, no matter how (x,y) approaches (0,0) for the limit of f(x,y) to exist. Any single path may or may not have a limit if f(x,y) itself doesn't have a limit.
     
  10. Oct 3, 2009 #9
    Re: Application of a Limit of a multivariable function..HELP!!

    how do I prove that with this paticular problem then?
     
  11. Oct 3, 2009 #10

    Dick

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    Re: Application of a Limit of a multivariable function..HELP!!

    You can't prove it because the limit of f(x,y) as (x,y) approaches (0,0) doesn't exist. You aren't supposed to do that. You are supposed to find a path along which f(x,y) approaches a. I'm trying to suggest you can find a path along which f(x,y)=a. I found some points approaching (0,0) where f(x,y)=2. What's the curve along which f(x,y)=2? Wouldn't that be a good path to choose for the case a=2?
     
  12. Oct 3, 2009 #11
    Re: Application of a Limit of a multivariable function..HELP!!

    so I took some values around zero of this function (N/D is @ zero)...and I see what you're saying..about the limit not exsisting:

    -0.05 0 -0.05
    0 N/D 0
    0.05 0 0.05

    but if we let f(x,y) = a = or equal say, z ... then we can show that the limit is a, for ANY a, as (x,y) is approching (0,0) ...I can see what you're saying now, but how to show that as a mathematical proof for ANY a? is where I'm confused.
     
  13. Oct 3, 2009 #12

    Dick

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    Re: Application of a Limit of a multivariable function..HELP!!

    Write f(x,y)=a and solve the equation for x. Just do it. What do you get?
     
  14. Oct 3, 2009 #13
    Re: Application of a Limit of a multivariable function..HELP!!

    I think I'm getting there...tell me if this makes sense to you:

    if we let the paths be: x=y and y=x:

    then the equations reduce to
    f(x,y=x) = x/2
    f(x=y,y) = y/2

    taking the limits of these two equations
    x-->a gives a/2
    y-->a gives a/2
    a/2=a/2 = a

    then the limit is a along this path as f(x,y)--> (0,0)
     
  15. Oct 3, 2009 #14

    Dick

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    Re: Application of a Limit of a multivariable function..HELP!!

    Nope, no sense whatsoever. Why are you ignoring my advice?
     
  16. Oct 3, 2009 #15
    Re: Application of a Limit of a multivariable function..HELP!!

    how would you express this equation explicitly in x?
     
  17. Oct 3, 2009 #16

    Dick

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    Re: Application of a Limit of a multivariable function..HELP!!

    What equation? Do you mean f(x,y)=a?? Write it out. It reduces to a linear equation to solve for x. Just TRY it.
     
  18. Oct 3, 2009 #17
    Re: Application of a Limit of a multivariable function..HELP!!

    2 = xy^2/(x^2+y^2)

    is this what you're talking about? how do I isolate x?
     
  19. Oct 4, 2009 #18

    Dick

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    Re: Application of a Limit of a multivariable function..HELP!!

    No it's not. Your original problem was f(x,y)=xy^2/(x+y^2). Start by multiplying both sides by the denominator.
     
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