Heya guys. I have to study the continuity of the following function at (0,0):
Thus I need to check whether
exists or not.
I converted to polar coordinates:
Thus the limit seems to be 0 except where
If , then
I then used L'Hopital's rule:
since .
However, by using such paths as and , the limit equals 1 and 2, respectively. What's going on that I don't understand here?
Thanks!
Also see the following graphic.
..........The numerator, Absolute value of the numerator, The denominator
Codes for Mathematica
Can you see that the (absolute value of) numerator is closer to than the denominator?Code:Plot3D[{Abs[x^3y],x^3y,x^4+y^2},{x,-1/2,1/2},{y,-1/2,1/2},PlotStyle->{{Red},{Yellow},{Blue}}]
So this implies that the limit must be at .
To prove this mathematically, you can lead the direction Opalg points.