# Finding a limit using polar co-ordinates.

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.

• You're right that $\theta$ can't be treated as a constant. See for example this recent answer (or many others on this site). – 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 $$0\leq \theta \leq 2\pi$$.
• 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) – Infinite Oct 30 '18 at 9:52