Michael Booth
Homework Week 2
Problem 24

ContourPlot[(((x*(y^3)))/(((x^2)+(y^6)))),{x,-0.001,0.001},{y,-0.001,0.001},PlotRange->All]

From this graph it looks as if the origin has a single elevation level, but this isn’t true. I’ve come to find that mathematica can actually “lie” to me. On a small enough scale mathematica cannot be trusted. So although this countour plot looks as though there is a single elevation level running through the origin this isn’t actually so.

ContourPlot[(((x*(y^3)))/(((x^2)+(y^6)))),{x,-1,1},{y,-1,1},PlotRange->All]

This contour plot shows different elevations meeting together. I believe these elevations actually meet at the origin, although it is not exactly obvious from the graph. As stated before mathematica cannot completely be trusted. If I use this contour plot along with my reasoning I can make an educated guess that these elevations do indeed meet at the origin, which is a problem.

Plot3D[(((x*(y^3)))/(((x^2)+(y^6)))),{x,-0.00001,0.00001},{y,-0.00001,0.00001},PlotRange->{-0.00001,0.00001},AxesLabel->{x,y}]

This graph is on a very small scale and shouldn’t be trusted, but I thought that it had some truth to it. If it actually does, I don’t know. From this graph I could argue that the limit does not exist.

Plot3D[(((x*(y^3)))/(((x^2)+(y^6)))),{x,-1,1},{y,-1,1},PlotRange->{-1,1},AxesLabel->{x,y}]

From this graph this looks like a continuous function. I’ve noticed that when the scale is large enough most functions do look continuous. Does that mean that they are? No. These graphs should be used along with reasoning to make judgements. 

The second contour plot, in the end, was the most useful in my eyes. I appreciate your help with this troublesome question. What I’ve taken from this is that mathematica isn’t perfect. Thanks for that insight. I honestly thought prior that these graphs were spot on, but they aren’t and that’s really good to know.

I’ll just tack a few things on to Michael’s file, so that you can see what he had to say about some of the visualization issues he’s encountered, and his conclusions

#5, p. 923: I’ll do this as an iterated limit. This turns out to provide the right answer in this (continuous) case.

Limit[Limit[5x^3-x^2y^2,x->1],y->2]

#10: trouble -- two different limits as we approach the origin!

Limit[Limit[5y^4Cos[x]^2/(x^4+y^4),x->0],y->0]
Limit[Limit[5y^4Cos[x]^2/(x^4+y^4),y->0],x->0]

#14: no trouble -- algebra to the rescue!

Simplify[(x^4-y^4)/(x^2+y^2)]