Why is it possible to calculate multivariable limits using polar coordinates? Let's say I'm looking for some $\lim_{(x,y) \to (0,0)}$ and I'm substituting $x = r cos\theta$ and $y = rsin\theta$ so that I can look at $\lim_{r \to 0}$.
Why can I do this? Am I not just looking at "straight lines" going to $(0,0)$ now? What about all the other possible sequences that converging in straight lines to (0,0)?
Why is it possible to calculate multivariable limits using polar coordinates
calculusmultivariable-calculusreal-analysis
Best Answer
While $r$ is going to $0$, $\theta$ is arbitrary. So, $\theta$ can freely change however it wants, as long as the radius is going to zero (that is, the convergence is uniform in $\theta$).
EDIT: See the following link for rigorous details: Polar coordinates for the evaluating limits