Geometrical significance

Question

What is the geometric significance of substitution? For instance substituting $x = r\cos(\theta)$ and $y = r\sin(\theta)$ in the following problem to find the limit. $\lim\limits_{(x,y) \to (0,0)} \dfrac{\tan(x^2 + y^2)}{x^2+y^2}$. Of course, this manipulation/substitution makes it easier to solve, but geometrically what does the above substitution mean. I hope the question is clear. Thank you in advance.

Answer 1

I see the geometrical interpretation as follows:

As the definition of limit requires the convergence for any sequence, the change to polar coordinates parametrizes in an easy way all linear (in the sense, along a line) approaches to the point, the angle parametrizes the slope at which we approach the point.