First, I'll give my definition of polar coordinates.
Let $f:U\subseteq\mathbb{R}^2\to\mathbb{R}$. Changing to polar coordinates we get $g:V\subseteq\mathbb{R}^2\to\mathbb{R},g(r,\theta)=f(r\cos\theta,r\sin\theta)$, where $r>0$ and $\theta\in[0,2\pi)$.
Now my question is, why is this true:
$$\lim_{(x,y)\to(0,0)} f(x,y)=L \Rightarrow \lim_{r\to0^+} g(r,\theta)=L $$.
But this isn't:
$$\lim_{r\to0^+} g(r,\theta)=L\Rightarrow\lim_{(x,y)\to(0,0)} f(x,y)=L$$
I've seen various counterexamples of this last statement so I know it is false, however I can't really understand why it doesn't work, so any help will be appreciated. Thanks.
Often polar coordinates are useful to handle limits in two variables and when the limit exists we have
$$\lim_{(x,y)\to(0,0)} f(x)=L \iff\lim_{(r\to 0^+)} g(r,\theta)=L$$
When limit doesn't exists we can obtain different limits for different paths but along a fixed path we obtain the same limit both in polar and cartesian coordinates.