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?


By substituting $x=r\cos\theta, y=r\sin\theta$ in the formula $f(x,y,z)$, you are not converting to "polar coordinates". A polar coordinate system is a two dimensional coordinate system by definition of the term.

Then what are you doing?
Well, the conversion you made, yields a system of coordinates that is known as a cylindrical coordinate system.

Why do we convert to polar coodinates sometimes?
Because $(x,y)\to (0,0)\iff r\to 0$, assuming the canonical conversion. This can make things easier, because now we only have to consider one variable $r$ in stead of two variables $x$ and $y$. However, mind that $\lim_{r\to0}$ needs to be treated with care. See this, this and this for instance.

Did I do something wrong?
Well, not yet. The substitution you made isn't wrong, is just not necessarily useful. If you convert to cylindrical coordinates and let $r\to0$, then you are not approaching the point $(0,0)$ but the $z$-axis. So if you were to continue using this method, you would have to calculate $\lim_{(r,z)\to(0,0)}$ (also a tricky thing). Because only then are you approaching $(0,0)$.

Is there a three dimensional equivalent of the polar coordinate system?
Yes, there is. It's called the spherical coordinate system. Once you've converted from cartesian coordinates to spherical coordinates, we have that $(x,y,z)\to(0,0,0) \iff r\to 0$. Once again, it will suffice to consider only one variable $r$ now, if we're lucky (mind that $\lim_{r\to0}$ is still a tricky thing).

I did not perform any calculations on your limit. I'm leaving that as an exercise to you. I would like to leave you with the note that converting to spherical coordinate isn't a magic way to solve any $\lim_{(x,y,z)\to(0,0,0)}f(x,y,z)$ problem, The same holds for polar coordinates, or any coordinate transformation for that matter. If it works, it works. If it doesn't, too bad.


The easy way to show that a limit does not exist is to show that we get two different answers approaching (0, 0, 0) in different directions. In each case, let u -> 0 Case 1 $x = u$; $y = u$; $z=\sqrt u$ Substituting, we get $3u^2/3u^2 = 1$ Case 2 $x = u$; $y = -u$; $z = \sqrt u$ Substituting, we get $u^2/3u^2 = 1/3$ Two different limits, so no limit.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.