How Can I Prove Uniform Convergence of This Function as ρ→0?

[malachia]

Probably is a silly question, but how could I prove that the function (expressed in polar coordinates)

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

converges to 0 as rho->0 uniformely in theta (if it is true, of course)?

 

[mathman]

It should be straightforward enough, since |sin|≤ 1 and the angle domain is a finite interval.

 

[Skrew]

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.