Question: Describe the singular behavior of: z=1/(x-y^2)

Answer: Using the 3D Graphing Calculator, you get a two surfaces with jagged edges, The jagged edges on the blue surface are pointing up while the jagged edges on the green surface are pointing down. The graph from the perspective of the 3D Level Surface Graphing Calculator clarifies the picture and indicates that the surface is asymptotically approaching the cylinder x=y^2.

Question: Describe the singular behavior of: z=y/x^2

Answer: Using the 3D Graphing Calculator, you get a two surfaces with jagged edges, The red jagged edges are pointing up while the yellow-orange jagged edges are pointing down. Two graphs from the perspective of the 3D Level Surface Graphing Calculator help clarify the picture and indicate that the surface is asymptotically approaching the plane x=0.

Note: The algorithm for creating 3d graphs may not give exactly what you expect and you may need to explore various ways to get what you want. In the case at hand, due to the singularity at y=0, it was necessary to get two graphs to fully see what is going on.

Question: Describe the singular behavior of: z=1/(x^2+y^2)

Answer: Using the 3D Graphing Calculator, you get a surface with jagged edges at the top, indicating asymptotic behavior. The graph from the perspective of the 3D Level Surface Graphing Calculator clarifies the picture and indicates that the surface is asymptotically approaching the z-axis.