Can polar coordinates always be used to calculate a limit in a multivariable function? Are polar coordinates always a viable way to calculate the limit of a multivariable function? 
In lecture, it appeared as if converting a function into polar coordinates and then checking the limit as r approaches 0 would be a foolproof way to determine a limit. However, after doing some online reading it appears as if it is not a viable method when the function is not "independently bound of theta". Could someone please explain this to me? I am having difficulty understanding this concept. 

If your function is $\frac{x}{x^2+y^2}$, then the polar substitution makes the denominator $r^2(\cos^2\theta+\sin^2\theta)=r^2$.
If your function is $\frac{x}{x^2+2y^2}$, then the polar substitution makes the denominator $r^2(\cos^2\theta+2\sin^2\theta)$, from which you can't just eliminate $\theta$ the same way.
