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Homework Help: A limit with two variables

  1. Apr 10, 2007 #1
    1. The problem statement, all variables and given/known data

    f(x, y) =((x^2)*y)/(x^4 + y^2) if (x, y) != (0, 0) ,
    f(x,y) = 0 if (x, y) = (0, 0) .
    a) Is f continuous at (0, 0)? Prove your statement.
    b) Show that
    -1/2 ≤ f(x, y) ≤1/2
    for all (x, y).

    I have used the two path test to show that it has not limit at 0,0 hence it is not continuous there. however, ı have no idea what I should do for the b part?Is there a algebratic way to show it or should we take differential etc.?
  2. jcsd
  3. Apr 10, 2007 #2
    Try solving for each inequality separately, and cross multiply to get a perfect square.
  4. Apr 10, 2007 #3
    Are you refering to the fx and fy (the partial derivatives of f) by saying "for each inequality"?
  5. Apr 10, 2007 #4
    What he means is to find necessary and sufficient conditions on x,y for each of the two inequalities

    [tex]-1 \leq \frac{2x^2y}{x^4+y^2}[/tex]


    [tex]1 \geq \frac{2x^2y}{x^4+y^2}[/tex]

    (ie. rearrange them, using "reversible" steps, until you find something that will tell you for which x,y they are satisfied)
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