Extra Example 1.  Show that the polynomial  [Graphics:Images/ComplexFunLimitMod_gr_222.gif]  is continuous at the point  [Graphics:Images/ComplexFunLimitMod_gr_223.gif]  in the complex plane.

Explore Solution for Extra Example 1.

Enter the function  [Graphics:../Images/ComplexFunLimitMod_gr_224.gif]  and find the limit.

[Graphics:../Images/ComplexFunLimitMod_gr_225.gif]

[Graphics:../Images/ComplexFunLimitMod_gr_226.gif]

 

 

 

We can use Mathematica to make a mapping of a neighborhood of  [Graphics:../Images/ComplexFunLimitMod_gr_227.gif]  in the z-plane and its image in the w-plane.

[Graphics:../Images/ComplexFunLimitMod_gr_228.gif]

[Graphics:../Images/ComplexFunLimitMod_gr_229.gif]

[Graphics:../Images/ComplexFunLimitMod_gr_230.gif]

[Graphics:../Images/ComplexFunLimitMod_gr_231.gif]

[Graphics:../Images/ComplexFunLimitMod_gr_232.gif]

We see that  [Graphics:../Images/ComplexFunLimitMod_gr_233.gif].  

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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