Honors Calculus IV

Limits and continuity

f (x, y) = ((8  (x^2) (y^2))/((x^4) + (y^4)))

If we approach the origin along a straight line, we find a limiting value, but different lines give different limits.

g1 := 
      Plot3D[8  x^2  y^2/(x^4 + y^4), {x, -2, 2}, {y, -2, 2}, 
        BoxRatios -> {1, 1, 1}, PlotPoints -> 50]

Show[g1, AxesLabel -> {"x", "y", "z"}, 
      AxesEdge -> {{-1, -1}, {1, -1}, {1, 1}}, Ticks -> Automatic]

f (x, y) = ((y^8)/((x^5) + (xy^6)))

\(TextForm
    \`Straight\ line\ approaches\ to\ the\ origin\ always\ yield\ the\ 
      limiting\ value\ zero, \ 
    but\ approach\ along\ the\ parabola\ x = 
      y\^2\ gives\ the\ limiting\ value\ \(one . \)\)

g2 := 
      Plot3D[y^8/(x^5 + x*y^6), {x, -2, 2}, {y, -2, 2}, 
        BoxRatios -> {1, 1, 1/2}, PlotRange -> {-1, 1}, PlotPoints -> 50]

Show[g2, AxesLabel -> {"x", "y", "z"}, 
      AxesEdge -> {{-1, -1}, {1, -1}, {1, 1}}, Ticks -> Automatic]