1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

2.3 • L I M ITS AND CONTINUITY 77

One technique for computing limits is to apply Theorem 2.4 to quotients. If
we let P and Q be polynomials and if Q (zo) f 0, then

lim P (z) = P (zo)
•-•o Q (z) Q (zo)'

Another technique involves factoring polynomials. If both P (zo) = 0 and
Q (zo) = 0, then P and Q can be factored as P (z) = (z - zo) P1 (z) and Q (z) =
(z -zo) Q1 (z). If Q1 (zo) f 0, then the limit is

lim P (z) = lim (z -zo) Pi (z) = P1 (zo).

•- •o Q (z) •-•o (z - zo) Q1 (z) Q1 (zo)

• EXAMPLE 2.19 Show that z-lli.m + t. z /~2z^2! 2 = 1 - i.

Solution Here P and Q can be factored in the form

P (z) = (z - 1 -i) (z + 1 + i) and Q (z) = (z -1 -i) (z - 1 + i)

so that the limit is obtained by the calculation

2 2·

1

. z - t li
1m = m
z-l+i z2 - 2z + 2 • -l+i

= lim

(z - 1 -i) (z + 1 + i)

(z - 1 -i) (z - 1 + i)

z +l+i

• -Hi z-l+i


• (1+i)+1 + i


• (1 + i) - 1 + i
  2 + 2i
  =--
  2i
  = 1 -i.

-------~EXERCISES FOR SECTION 2.3

1. Find the following limits.

(a.) lirn (z^2 - 4z + 2 + 5i).

z-2 + i

(b) z_.t lim z^2 ±.t:+l 4zt2.

(c) ,;Lim -i .. ,;--. 1 1.