Expression 21: "f" left parenthesis, "x" , "y" , right parenthesis equals 1 plus StartFraction, "x" "y" Over "x" squared plus "y" squared , EndFractionfx,y=1+xyx2+y2
21
Set surfaces (wrench icon) to TRANSLUCENT.
22
To examine the limit of z=f(x,y) along a path in the xy-plane, LIFT the path, staying parallel to the z-axis, and LIE the path on the surface y=f(x,y).
23
When y=mx, the z=f(x,y) is:
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Expression 25: to "z" equals "f" left parenthesis, "x" , "m" "x" , right parenthesis equals 1 plus StartFraction, "m" Over "m" squared plus 1 , EndFraction equals StartFraction, "m" squared plus "m" plus 1 Over "m" squared plus 1 , EndFraction→z=fx,mx=1+mm2+1=m2+m+1m2+1
25
Approaching (0,0) along the (orange) path y=x (so y=mx with m=1).
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26
Approaching (0,0) along the (purple) path y= -x (so y=mx with m= -1).
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43
Approaching (0,0) along the (green) path y=0 (so y=mx with m=0)
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55
Approaching (0,0) along the (pink) path y=mx for m between -1 and 1.
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72
Approaching (0,0) along the (teal) path y=5x^3 (a cubic)
