Quick Qns : Convergence in Polar Coordinates

[Legendre]

Suppose we investigating the limit of a function on R^2 as (x,y) tend to (0,0).

We convert the function into polar coordinates.

Then "(x,y) tend to (0,0)" is equivalent to "r tend to 0"?

Theta (the angle) does not matter?

 

[Hootenanny]

Suppose we investigating the limit of a function on R^2 as (x,y) tend to (0,0).

We convert the function into polar coordinates.

Then "(x,y) tend to (0,0)" is equivalent to "r tend to 0"?

Theta (the angle) does not matter?

Yes, the angle theta certainly matters. As no doubt you know, a limit it R² only exists if the value of the limit is independent of the path along which the limit is taken. You would also need to account for theta if, for example, you were asked to compute the limit along a certain path.

 

[HallsofIvy]

But if the limit itself exists, the fact that it is independent of the path means that the angle does not matter.

 

[Legendre]

Thanks for the input~