The discussion revolves around evaluating the limit of the function ln(x^2 + y^2) as (x, y) approaches (0, 0). Participants explore different approaches to understand the behavior of the function near this point.
Participants are actively exploring various methods to approach the limit, with some discussing the implications of using polar coordinates to simplify the problem. There is also a side discussion about a different limit involving three variables, where multiple approaches yield different results, prompting questions about the validity of those methods.
Some participants express uncertainty about the legality of setting variables equal to specific functions of another variable while evaluating limits. There is also mention of differing methods presented in textbooks, indicating a variety of approaches to limit evaluation in multivariable calculus.
Fightfish said:The form (x^2+y^2) strongly suggests a parameterisation, yes?
Fightfish said:Ah, so you get to learn a new technique today:).
Let's use polar coordinates:
x = r cos \theta, y = r sin \theta
Then our limit becomes
\lim_{r \to 0} ln (r^2) = 2 \lim_{r \to 0} ln (r)You realize here that we have managed to remove the dependence on the direction of approach - ie our answer is independent of \theta.
Yup. You can choose to approach the limit from any direction that you like. In your second case, you were approaching it from a parabolic path; that's fine.theBEAST said:What I did was I set x,y = 0 and solved the limit as z -> 0 which gave me zero.
Next I set y=z^2 and x=z^2 and found the limit as z->0 which gave me 1. Therefore the limit does not exist.
Is this a correct method? I am not sure if it is legal to independently set y and z equal to z^2.
theBEAST said:PS: the textbook did it differently they set y=x and z=0 and let the limit x -> 0 but I think there should be multiple ways to do this.