Section 15.3: Limits and Continuity

> restart;

> with( plots ): 

Warning, the name changecoords has been redefined

> setoptions3d( axes=boxed, shading=zhue );

>

Example 1

> f := (x^2-y^2)/(x^2+y^2);

f := (x^2-y^2)/(x^2+y^2)

> plot3d( f, x=-2..2, y=-2..2 );

[Maple Plot]

Rotate this plot to see the strange behavior at the origin. While there should be a "hole" at the origin, Maple "connects the dots".

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> plot3d( f, x=-2..2, y=-2..2, style=contour, orientation=[-90,0] );

[Maple Plot]

Note that every contour is a straight line through the origin. (Technically, the origin is NOT on any level curve but, once again, Maple connects the dots.)

For any delta >0 there are points on any of the level curves that are within delta of the origin. This fact PROVES this function has no limit at the origin. This fact can also be seen by rotating the contour plot to one of the side views.

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> plot3d( f, x=-2..2, y=-2..2, style=contour, orientation=[-90,90] );

[Maple Plot]

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An analytic proof that the limit at the origin does not exist can be based on the observation that, when evaluated along the line through the origin with slope m, the function reduces to

> eval( f, y=m*x );

(x^2-m^2*x^2)/(x^2+m^2*x^2)

> f_m := simplify( % );

f_m := -(-1+m^2)/(1+m^2)

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Because this expression depends on the slope, it is clear that the value of the function along lines whose slopes have different absolute values are different.

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