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Two variable limits

  1. Mar 31, 2009 #1
    1. The problem statement, all variables and given/known data

    The real valued function f of two variables is defined by f(x,y) =
    tan( (1/2) *(pi) *sqrt( (x^2 + y^2) )/( sqrt( (x^2 + y^2) ) for each (x,y) satisfying 0 < x^2 + y^2 < 1.

    How should f(0,0) defined so that f is continuous at (0,0)?

    2. Relevant equations



    3. The attempt at a solution
    I wasn't sure whether the question was asking show the limit exists or doesn't exist? I think asking me to show it exists.
    I tried using polar co ordinates
    y=r*sin[tex]\theta[/tex] , x=r*cos[tex]\theta[/tex]
    =tan( (1/2) * (pi) * r) / r
    = 1/cos( (1/2) * (pi) * r ) * sin((1/2) * (pi) * r)/r
    = 1 * sin((1/2) * (pi) * r)/r

    I got stuck here was trying to use the identity lim->0 sinx/x = 1
    I know you can use squeeze theorem to prove limits exist, but I don't really understand how it works for two variables.
    In one variable limits if 0 < |x-a|<[tex]\delta[/tex] then |f(x)-L|< [tex]\epsilon[/tex]
    but in two variable limits what is |x-a|?
     
  2. jcsd
  3. Apr 1, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    since sin(x)/x-> 1, for [itex]sin(\pi r/2)/r, let [/itex]x= (\pi/2)r[itex]. What is r equal to in terms of x? Replace the r in the denominator by that.
     
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