0
$\begingroup$

Let us suppose that we have to find the following limit: $$\lim_{(x,y)\to (0,0)} f(x,y).$$ Can we solve such a limit using polar co-ordinates? I have seen the following method somewhere on the internet:

If we take $x=r\cos\theta,\ y=r\sin\theta$, then the above limit becomes:$$\lim_{r\to 0}f(r\cos\theta,r \sin\theta).$$ But solving a limit this way does not cover all the paths passing through $(0,0)$, because whatever $\theta$ we choose, that $\theta$ gives a straight line path through $(0,0)$, it does cover the path like $x^2, \ x^3$ etc.

$\endgroup$
  • $\begingroup$ You're right that $\theta$ can't be treated as a constant. See for example this recent answer (or many others on this site). $\endgroup$ – Hans Lundmark Oct 30 '18 at 9:45
1
$\begingroup$

To prove that the limit is $L$ you have to show that $f(r\cos \theta,r\sin \theta)\to L$ as $r \to 0$ uniformly for $0\leq \theta \leq 2\pi$.

$\endgroup$
  • $\begingroup$ Uniformly means, for all $\theta$, $f$ tends to $L$. Am I right? (But my question is: whatever $\theta$ we choose, it gives a straight line path only, it does not cover all the paths) $\endgroup$ – Infinite Oct 30 '18 at 9:52

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.