Question

prove limit does not exist with polar coordinates.

lim (x,y) => (0,0) [(x^2y)/(x^2 + y^2)^(3/2)

Answer 1

x^2y = (r^2 cos^2)(r sin) = r^3 cos^2 sin

(x^2 + y^2)^3/2 = (r^2)^3/2 = r^3

\[\lim_{(xy) \rightarrow (0,0)} \cos^2 \theta \sin \theta\]
not quite sure where to go from here

Answer 2

as (x,y) --> (0,0) this is the same as saying r --> 0

Answer 3

I think the fact that theta being on the right side makes this limit not equal to some fixed number, which points to the limit not existing. Not sure though

Answer 4

could be .... tan theta = y/x = 0/0 so theta is indeterminate
so limit is indeterminate