11.2 - limits - WebAssign problems...

Next to last problem

myf[x_,y_]:=x^2 y E^y/(x^4+7y^2)

ParametricPlot3D[{x,y,myf[x,y]},{x, -.01,.01},{y,-.01,.01}, AxesLabel->{x,y,z}]

For the limit to exist, any path heading towards (0,0) must have the same limit. 
What about approaching (0,0) along the line y=0? 

Try plotting h(y)=f(.1,y)h(y)=f(0.1,y):

Plot[myf[0.1,y],{y,-1,1}]

Last problem (polar coordinates)

How to evaluate (lim_((x,y)→(0,0)))((x^2)+(y^2))ln((x^2)+(y^2))?  

Converting to polar coordinates, (x^2)+(y^2)=(r^2):

({{({{(lim_((x,y)→(0,0)))((x^2)+(y^2))ln((x^2)+(y^2))=(lim_(r→(SuperPlus[0]), any θ))((r^2))ln((r^2))}})}})

This suggests plotting f(r)=(r^2)ln((r^2)).   Try it for 0≤r⩽2 !

In polar coordinates, you only need to consider positive values of r, since we can reach all the points in the x-y plane with positive r values, and 0≤θ<2π.