Functions. Limits and Continuity. Arcs and CurvesthatArgL-a < Argf(z) < ArgL+a
orI Argf(z) -ArgLI <a
whenever z E N6( a) n D. But since
/3 = ILi sin a::; ILi sin E^1
it follows that a ::; E^1 ::; E, so (3.7-2) givesI Argf(z) -ArgLj < E whenever z E NHa) n D
137(3.7-2)which shows that limz-+a Arg f ( z) = Arg L. A similar result holds for any
other particular determination of the argument.
3.8 Infinite Limit at a Finite Point
Definition 3.3 Let f: D --? <C be a single-valued complex function, and
let a (a f =)be an accumulation point of D. We say that f(z) has an
infinite limit as z approaches a [or that = is the limit off ( z) at a] if for
every K > 0 there is a 8 > 0 (depending on K) such that
lf(z)j > K whenever 0 < jz - al< ·5 and z ED
or alternatively, if
z E NHa) nD implies jf(z)I > K or XCf(z), CXJ) < E
where E = 1/(1 + K^2 )^112 • This type of limit is denoted briefly by writing
lim f(z) = CXJ
z-+aor f(z)--? oo asz--?a
As before, the limit so defined is necessarily unique.
Example Let f(z) = l/(z-3), D = <C-{3}. Then we have limz_, 3 f(z) =
CXJ. In fact, for any given K > 0, the inequality l/lz - 31 > K holds
whenever
10 < jz - 3j < K = 8
Theorem 3.3 Let f: D --? <C, g: D --? <C, and a finite, an accumulation
point of D. Suppose that limz-+a f(z) = CXJ and limz-+a g(z) = L f CXJ.
Then we have
1. limz-+aCf ± g)(z) = CXJ
- limz-+a(fg)(z) = CXJ