I have a doubt about finding the limit of two variable functions $f(x,y)$ at some general point say $(a,b)$ without loss of generality $a >0, b>0.$
Polar Coordinates as a Definitive Technique for Evaluating Limits
https://en.wikipedia.org/wiki/Polar_coordinate_system
The relation between cartesian coordinates and polar coordinates is as
$$x=r\cos(\theta), y=r\sin(\theta)$$ where $r \geq 0$ and $\theta \in (0, 2\pi].$
$$r=\sqrt{x^2+y^2}, \theta = \tan^{-1}\frac{y}{x}$$
Doubt: Do these two expressions are equivalent to finding the limit of $f(x,y)$?
$$ \lim_{(x,y)\to(a,b)}f(x,y)=\lim_{(r,\theta) \to (\sqrt {a^2+b^2},\theta)}f(r,\theta) -----(*)$$
where on the right-hand side in $(*)$, we fixed $r$ and then checked expression depends on $\theta $ or not we determine whether limit exists or not.
For instance limit at origin that is $(x,y) \to (0,0)$ then $(r,\theta) \to (0,\theta)$.
I'm a little confused about this concept. Please correct me if I'm wrong somewhere.
Thanks in advance.
Edit: For example
$$\lim_{(x,y) \to (0,0)} \frac{x^2+y^3}{x+y+1} = 0$$
Using polar coordinates it is easily seen that $$\lim_{(r,\theta) \to (0,\theta)} r^2[\cos^2 \theta+r\sin^3 \theta/r\cos \theta+r\sin \theta+1]=0$$