Evaluate limit approaching (a,b) using polar coordinates

Question

I know how to use polar coordinates to evaluate a limit where $(x,y)$ approaches $(0,0)$. But how do I do it if $(x,y)$ approaches a different point than $(0,0)$?

I thought of perhaps still using that $(r \to 0)$ but when I translate the function use that

$r^2=(x-a)^2 + (y-b)^2$

$x = r \cos(\theta) + a$

$y = r \sin(\theta) + b$

(Assuming that $(x,y)$ approaches $(a,b)$)

Would that give me a correct answer?

Answer 1

We can simply take $(x,y)=(u+a,v+b)$ with $(u,v)\to (0,0)$ and then use polar coordinates for $u$ and $v$ that is precisely equivalent to your way.

Answer 2

Yes, your idea of $$ x=a+ r\cos \theta, y=b+r\sin \theta $$ works fine.

Go for it.