Lecture 12 Complex Analysis

2.3 Limits:-
Let f(Z) be defined and single-valued in a neighborhood of Z_0(It is the set of all points Z such that |Z-Z_0 |?? where ? is positive real number) with the possible exception of Z_0 it self. We say that the number l is the limit of f(Z) as Z approaches Z_0 and write
lim?(Z?Z_0 )??f(Z)=? l. "If for any positive number ?, we can find some positive number ? (depending on ? )such that |f(Z)- l|<? whenever 0<|Z-Z_0 |< ? " .
In such case we also say that f(Z) approaches l as Z approaches Z_0 and write f(Z)?l as Z?Z_0. We can represent the above definition of limit by geometrical representation.
EX(1):- If f(Z)=1/2 iZ is defined
on |Z|<1 prove that


lim?(Z?1)??iZ/2=? i/2
by definition
Sol:- We see that 1 isn t belong to the
domain of f [|Z|<1]. Let ?>0, then
|f(Z)-l|<??|1/2 iZ-i/2|<??|i/2(Z-1)|<?
?|i/2||Z-1|<??1/2 |Z-1|<?
?|Z-1|<2??|Z-1|<? where ?=2?
Thus ??>0 ?|1/2 iZ-i/2|<? ? ?=2?? |Z-1|<?
EX(2):- Prove that lim?(Z?(2i-1) )??(2Z+3)=4i+1? by def.
Sol:- Let ?>0, then |2Z-4i+2|<?
|(2Z+3)-(4i+1) |<??|2(Z-(2i-1)) |<?
?|2||Z- (2i-1) |<??|Z-(2i-1) |<?/2
EX(3):- If f(Z)={?(?(Z^(2 ) Z?i@ )@ 0 Z=i)? Prove that lim?(Z?i)??f(Z)=1 ?? H.W
Limit of ? :- If we put the function ?=f(Z) for all Z with 1/Z , we get a new function g(Z)=f(1/Z) . If Z?? , then 1/Z?0 to find the limit of function =f(Z) . That is when Z?? suffice to the limit of function g(Z)=f(1/Z) when Z?0.
EX:- Find lim?(Z??)??(Z-1)/(3Z+1)?
Sol:- Let f(Z)=(Z-1)/(3Z+1)
g(Z)=f(1/Z)=(1/Z-1)/3(1/Z+1) =((1-Z)/Z)/((3+Z)/Z)=(1-Z)/(3+Z)
lim?(Z?0)??g(Z)=lim?(Z?0)?? (1-Z)/(3+Z)=(1-0)/(3+0)? ?=1/3=lim?(Z??)??f(Z)? ?=lim??(Z??)??(Z-1)/(3z+1)?