Extra Example 3.  Show that the polynomial  [Graphics:Images/ComplexFunLimitMod_gr_249.gif]  is continuous at the point  [Graphics:Images/ComplexFunLimitMod_gr_250.gif]  in the complex plane.

Explore Extra Example 3.

Enter the function  [Graphics:../Images/ComplexFunLimitMod_gr_251.gif]  and find the limit.

[Graphics:../Images/ComplexFunLimitMod_gr_252.gif]

[Graphics:../Images/ComplexFunLimitMod_gr_253.gif]

 

 

We can use Mathematica to make a mapping of a neighborhood of  [Graphics:../Images/ComplexFunLimitMod_gr_254.gif]  in the z-plane and its image in the w-plane.

[Graphics:../Images/ComplexFunLimitMod_gr_255.gif]

[Graphics:../Images/ComplexFunLimitMod_gr_256.gif]

[Graphics:../Images/ComplexFunLimitMod_gr_257.gif]

[Graphics:../Images/ComplexFunLimitMod_gr_258.gif]

[Graphics:../Images/ComplexFunLimitMod_gr_259.gif]

We see that  [Graphics:../Images/ComplexFunLimitMod_gr_260.gif].  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2006 John H. Mathews, Russell W. Howell