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Question on uniform convergence

  1. Nov 10, 2011 #1
    Probably is a silly question, but how could I prove that the function (expressed in polar coordinates)

    \left(\rho^4\cos^2{\theta} + \sin^3{\theta}\right)^{\frac{1}{3}} - \sin{\theta}

    converges to 0 as rho->0 uniformely in theta (if it is true, of course)?
  2. jcsd
  3. Nov 10, 2011 #2


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    Science Advisor

    It should be straightforward enough, since |sin|≤ 1 and the angle domain is a finite interval.
  4. Nov 10, 2011 #3
    Ok well your functions are continuous. So show that inside goes to sin^3(theta), then the cubed root is going to equal sin(theta), then subtract to get 0.

    But this is assuming you have defined or can assume x^(1/3) is defined and is continuous.
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