Guess at a possible value for a lim z21 1z1z b lim z21 z22iz

Guess at a possible value for (a) lim z->2+1 (1-z)/(1+z), (b) lim z->2+1 z^2-2iz)/(z^2+4) and investigate the correctness of your guess

Solution

(a ) lim Z-> 2 +1i (1- Z) /(1 + Z)

= (1 - 2 -1i ) / ( 1 + 2 + 1i)

= (-1 - i) / ( 3 + 1i)

divide and multiple with (3 - i).

= [(-1-i) ( 3- i) ] / [ ( 3 + i) ( 3 - i) ]

= [ -2 - 2i ] / [ 3² - i ² ]

= - (1 + i) / 4

( b ) lim Z -> 2 +1i (Z^2 - 2 iZ ) / ( Z^2 +4 )

= ( 2 + 1i)^2 - 2i( 2 + 1i ) / ( (2 +i )^2 +4)

= ( 4 + i² + 4 i) - (4i + 2i² ) / ( 4 + i^2 +4i)

= 5 + 4i - 4 i - 2 ) / ( 5 + 4i )

= 3 / ( 5 + 4 i)

multify and divide by (5 - 4i)

= 3 * ( 5 -4 i) / ( 5-4i)* (5 + 4i)

= 3 (5 - 4i) / 9

= (5 - 4i) / 3