When working on limits of functions with two variables, $f(x,y)$, I like to convert the problem to polar coordinates a lot of the time, by changing the question from $$\lim_{(x,y)\to (0,0)}f(x,y)$$ to $$\displaystyle\lim_{r\to 0}f(r\cos\theta,r\sin\theta).$$ I was just doing some problems in my book when I encountered a limit of a function with three variables, $f(x,y,z)$. I was just wondering if there was a way to calculate such a limit with polar coordinates.
An example being: $$\lim_{(x,y,z)\to(0,0,0)}\frac{xy+yz^2+xz^2}{x^2+y^2+z^4}$$
Converting it into polar coordinates gives me:
$\displaystyle\lim_{r\to 0}\dfrac{r^2\sin\theta\cos\theta+r\sin\theta \cdot z^2+r\cos\theta\cdot z^2}{r^2(\sin^2\theta+\cos^2\theta)+z^4}=\displaystyle\lim_{r\to 0}\dfrac{r(r\sin\theta\cos\theta+\sin\theta\cdot z^2+\cos\theta\cdot z^2)}{r^2+z^4}$
Can I proceed or is polar coordinates strictly for use with two variables only?
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