xtra Example 2.  Show that the polynomial  [Graphics:Images/ComplexFunLimitMod_gr_237.gif]  is continuous at the point  [Graphics:Images/ComplexFunLimitMod_gr_238.gif]  in the complex plane.

Explore Solution for Extra Example 2.

Enter the function  [Graphics:../Images/ComplexFunLimitMod_gr_239.gif]  and find the limit.

[Graphics:../Images/ComplexFunLimitMod_gr_240.gif]

[Graphics:../Images/ComplexFunLimitMod_gr_241.gif]

 

 

 

We can use Mathematica to make a mapping of a neighborhood of  [Graphics:../Images/ComplexFunLimitMod_gr_242.gif]  in the z-plane and its image in the w-plane.

[Graphics:../Images/ComplexFunLimitMod_gr_243.gif]

[Graphics:../Images/ComplexFunLimitMod_gr_244.gif]

[Graphics:../Images/ComplexFunLimitMod_gr_245.gif]

[Graphics:../Images/ComplexFunLimitMod_gr_246.gif]

[Graphics:../Images/ComplexFunLimitMod_gr_247.gif]

We see that  [Graphics:../Images/ComplexFunLimitMod_gr_248.gif].  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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