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Limits if polar coordinates (conceptual explanation)

  1. Mar 29, 2014 #1
    1. The problem statement, all variables and given/known data[/b]

    Suppose the lim(x,y) →(0,0) (xy)/SQRT[x^2 + y^2] if it exists

    find the limit.



    3. The attempt at a solution

    x = rcosΘ
    y = r sinΘ
    r = SQRT[x^2 + y^2]

    ∴ limr → 0 (r2cosΘrsinΘ)/ r = rcosΘsinΘ [itex]\leq r[/itex]

    and so -r [itex]\leq(xy)/SQRT[x^2 + y^2][/itex] [itex]\leq r[/itex]

    ...
    ...

    I can understand the limit goes to zero because algerbraic multiplication and the sandwhich theorem tells me so. However, the highlighted part in red confuses me. What is the significance of
    rcosΘsinΘ [itex]\leqr[/itex]?
     
  2. jcsd
  3. Mar 29, 2014 #2

    UltrafastPED

    User Avatar
    Science Advisor
    Gold Member

    It was to generate the inequality - they used cosΘrsinΘ ≤ 1.
     
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