Intuition for limits computed using polar coordinates

Question

I would like to understand how can we change from multivariable limit to a limit that only involves r Because in school they tell me : $x = cos(\theta)$ $y = sin(\theta)$ $r^2 = x^2 +y^2$ so as $(x,y) \lim_{(x,y)\to(0,0)}$ we have $r\to 0^+$ I learn this and I just do it mechanical. But I would like to see why this is true. And somewhere, I have seen that if the function depends on $x$ and $y$, i.e. $f(x,y)$ then when doing the change of variable to polar coordinates we get $f(r, \theta)$. However, when writing down the limit in polar coordinates, we only put as $r\to 0^+$, where do we leave the theta and why? Why shouldn't it be as $(r,\theta)\to (a,b)$ where $a$, $b$ belong to $\Bbb R$? Thanks a lot !