138 Chapter^3- limz_,a(f /g)(z) = oo
4. limz-->a (g / !)( Z) = 0
- limz-->af(z) = 00
- limz->a IJ(z)I = 00
Proofs (1) By Theorem 3.2-8 we have limz-->a lg(z)I = ILi. Hence there
is a 01 > 0 such that z E N6 1 (a) n D implies that lg(z)I < ILi + 1. Bydefinition 3.3, for any given I< > 0 there is a Oz > 0 such that z E N6 2 (a )nD
implies that lf(z)I > ]{ + ILi + 1. Thus 8 = min(o1, Oz), z E N6(a) n D
implies. that
IJ(z) ± g(z)I ~ IJ(z)l - lg(z)I >I<
(2) If L i- 0, corresponding to E =^1 / 2 ILI there is o 1 > 0 such that
z E N6 1 (a) n D implies that
-^1 /2ILI < lg(z)I - ILi <^1 /2ILI
and hence lg(z)I >^1 / 2 ILI. Also, for any given ]{ > 0 there is Oz > 0
such that z E N6 2 (a) n D implies that IJ(z)I > 2I</ILI. Therefore, for
o = min(o1,0 2 ) and z E N8(a) n D, we have
IJ(z)g(z)I = IJ(z)ljg(z)I > ]{
The proofs of the remaining properties are left as exercises.Example Consider the function F(z) = 1/[(z - 2)(.i + 1)]. Let J(z) =
1/(z - 2) and g(z) = 1/(z + 1). Since
lim J(z) = oo
z-->2and limg(z) = 1/s i- 0
z-->2it follows, by property 2 above, that limz_. 2 F(z) = oo.
3.9 Infinite Limit at Infinity
Definition 3.4 Let f: D --; C be a single-valued complex function, and
suppose that oo is an accumulation point of D. The point oo is said to be
the limit of J(z) as z approaches oo [or the limit of f(z) at oo] if for every
]{ > 0 there is an M > 0 (depending on K) such that
IJ(z)I >I< whenever lzl > M and z ED
or alternatively, if
z E N6(oo) n D implies f(z) E N~(oo)
with 8 = 1/(1 + M^2 )^112 and E = 1/(1 + ]{2)1/2.