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.

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    $\begingroup$ 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

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 $0\leq \theta \leq 2\pi$.

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  • $\begingroup$ 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

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