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?
Convergence in polar coordinates refers to the behavior of a sequence of points in a polar coordinate system as the number of points increases towards infinity. It is a measure of how closely these points approach a specific limit or point of interest.
Convergence is important because it allows us to determine whether a series of points in a polar coordinate system will eventually reach a specific limit or point of interest. It helps us understand the behavior of a function or sequence and make predictions about its future behavior.
The convergence of a sequence of points in polar coordinates can be calculated by evaluating the limit of the distance between each point and the desired limit as the number of points increases towards infinity. If this limit is equal to zero, the sequence is said to converge.
If a sequence of points in polar coordinates does not converge, it means that the points do not approach a specific limit or point of interest. This could indicate that the function or sequence is oscillating or has no defined limit in the given range.
The main difference between convergence in polar and Cartesian coordinates is the way the distance between points is calculated. In polar coordinates, the distance is calculated using the polar distance formula, while in Cartesian coordinates, the distance is calculated using the Pythagorean theorem. Additionally, the behavior of a function or sequence in polar coordinates may be different due to the presence of an additional angle variable.