The discussion revolves around the concept of limits in the context of polar coordinates, specifically examining whether the convergence of a function as (x,y) approaches (0,0) can be simplified to considering only the radial component (r) while disregarding the angular component (theta).
Participants express differing views on the relevance of theta in the limit process, indicating that the discussion remains unresolved regarding the role of the angle in determining limits in polar coordinates.
The discussion highlights the dependence on the path taken to approach the limit, suggesting that assumptions about the behavior of the function may vary based on the chosen approach.
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.Legendre said: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?