Functions. Limits and Continuity. Arcs and Curves 135Example If f(z) = l/z^2 ,D = <C-{O}, then limz-+oof(z) = 0. In fact,
given E > 0 we haveby taking
1lzl > ,.fa= M
- 7 , PROPERTIES OF FINITE LIMITS
Theorem 3.2 Let f: D ----+ <C, g: D ----+ <C, and a (finite or oo) an accu-
mulation point of D. Suppose that limz-+a f ( z) = L and limz-+a g( z) = L',
with both L and L' finite. Then we have:
1. If f(z) = c (c-=/:-oo a constant) on N/;(a)nD for some 8 > 0, then L = c.
2. limz-+a(f ± g)(z) = L ± L'.
- limz-+a(fg)(z) = LL'.
4. limz-+a(f/g)(z) = L/L' proyided that L'-=/:-0.
5. limz-+a f(z) = L.
6. limz-+aRef(z) = ReL, limz-+almf(z) = ImL.
- Equations (6) imply that limz-+af(z) = ReL + ilmL = L.
8. limz-+a IJ(z)I = ILi..
9. limz-+aArgf(z) = ArgL provided that L + ILi-=/:- 0
Proofs (1) For any E > 0 and 0 < 8' :; 8 we have
lf(z) - cl= 0 < c: ,whenever z E N~, (a) n D
In the case a = oo, the notation N/;(a) is to be understood as a delet~d
neighborhood of oo in the chordal metric. The same remark applies in
what follows.
(2),(3),( 4) The proofs of these properties are formally the same as in
real analysis.
(5) By Definition 3.1 (or 3.2, as the case may be), for every c: > 0 there
is a 8 > 0 such that
lf(z) - LI< c:But this implies that
lf(z) - LI < E
Hence lim;:-+af(z) = L.
whenever z E N~(a) n D
whenever z E N~(a) n D