Here are graphs illustrating some examples of limits from the textbook, Section 14.2.

Please give the applets a few seconds to load. The resolution has been set higher (plot_points=200) so that we can see the behavior of the function close to the origin.

Variant of Example 1

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var('x y')

def f(x,y): return x^2/(x^2+y^2)

gr = plot3d(f(x,y), (x,-1,1), (y,-1,1), plot_points=200)

zmin = gr.bounding_box()[0][2]

zmax = gr.bounding_box()[1][2]

ax = text3d("x",(0,-1,zmin))

ay = text3d("y",(1,0,zmin))

az = text3d("z", (-1,-1,(zmin+zmax)/2) )

show(gr+ax+ay+az, axes=True, frame=True)

Example 2

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var('x y')

def f(x,y): return (x*y)/(x^2+y^2)

gr = plot3d(f(x,y), (x,-1,1), (y,-1,1), plot_points=200)

zmin = gr.bounding_box()[0][2]

zmax = gr.bounding_box()[1][2]

ax = text3d("x",(0,-1,zmin))

ay = text3d("y",(1,0,zmin))

az = text3d("z", (-1,-1,(zmin+zmax)/2) )

show(gr+ax+ay+az, axes=True, frame=True)

Example 3

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var('x y')

def f(x,y): return (x*y^2)/(x^2+y^4)

gr = plot3d(f(x,y), (x,-1,1), (y,-1,1), plot_points=200)

zmin = gr.bounding_box()[0][2]

zmax = gr.bounding_box()[1][2]

ax = text3d("x",(0,-1,zmin))

ay = text3d("y",(1,0,zmin))

az = text3d("z", (-1,-1,(zmin+zmax)/2) )

show(gr+ax+ay+az, axes=True, frame=True)

Variant of Example 4

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var('x y')

def f(x,y): return (x*y^2)/(x^2+y^2)

gr = plot3d(f(x,y), (x,-1,1), (y,-1,1), plot_points=200)

zmin = gr.bounding_box()[0][2]

zmax = gr.bounding_box()[1][2]

ax = text3d("x",(0,-1,zmin))

ay = text3d("y",(1,0,zmin))

az = text3d("z", (-1,-1,(zmin+zmax)/2) )

show(gr+ax+ay+az, axes=True, frame=True)