A couple more limit example from Section 14.2
#17, #15, p. 923
Andy Long, Fall, 2015

#15. The product of continuous functions, hence continuous on its domain. The question, however, is (0,0):

f[x_,y_]:=x ^2y/(x^4+4y^2)E^y
width=.01
Plot3D[f[x,y],{x,-width,width},{y,-width,width},AxesLabel->Automatic]
(* all parabolic paths y=mx^2 are troublesome! *)
(* What we see here are multiple contours trying to meet at a point! *)
ContourPlot[f[x,y],{x,-width,width},{y,-width,width}]

Plot[f[x,x^2],{x,-1,1},AxesOrigin->{0,0}]
Plot[f[x,.5*x^2],{x,-1,1},AxesOrigin->{0,0}]

#17: consider polar coordinates:

f[x_,y_]:=(x^2+ y^2)/(Sqrt[(x^2+ y^2)+1]-1)
width=.1 (* .0001 is way too cool! *)
Plot3D[f[x,y],{x,-width,width},{y,-width,width},AxesLabel->Automatic]
ContourPlot[f[x,y],{x,-width,width},{y,-width,width}]

g[x_,y_]:=(Sqrt[(x^2+ y^2)+1]+1)
width=.000001 (* .0001 is too boring! *)
Plot3D[g[x,y],{x,-width,width},{y,-width,width},AxesLabel->Automatic]
ContourPlot[g[x,y],{x,-width,width},{y,-width,width}]