limits

suppose f is defined at all points in some nbhd of a point z_0 ,by the statment
that
lim?(z?z_0 )??f(z)=w_0 …(*)?
we mean that w=f(z) can be made arbitrary close to w_0 provided z
is sufficiently close to z_0 and distinct from it.
mathematically:-
this means for each ?>0 arbitarly small there exist ?>0 ?
|f(z)-w_0 |i.e. ???(?,zo).
Geometrically:-
this means for each ?-nbhd |w-w_0 |?-nbhd of z_0 such that each point of this a deleted ?-nbhd of z_0 such that each point of this nbhd has an image lying in the ?-nbhd of w_0 since (*)holds for each point of the nbhd we have
z goes to z_0 in any path .
f should be define at all point
there must be the connectedness
notice that nbhd of z_0 comes into enistance if z_0 is an interior point .
(in particular in the case of an open set)
by the uniform limit we mean ? depends only on ?.(i.e independent of z_0 )
uniqueness of the limit?-
suppose lim?(z?z_0 )??f(z)=w_0 ,? lim?(z?z_0 )??f(z)=w_1 ,?
?w_0=w_1
theorem:-
let f(z)=u+iv ,z_0=x_0+?iy?_(0 ,) w_0=u_0+?iv?_0
then
lim?(z?z_0 )??f(z)=w_0 if and only if lim?((x,y)?(x_0,y_0))??u(x,y)=u_0 ? ?
and
lim?((x,y)?(x_0,y_0))?v(x,y)??=v_0 ?
remarks?-
lim?(z?z_0 )??f(z)=w_(0 ) ? lim?(z?z_0 )??(f(z) ) ?=(w_0 ) ? ? ?
let c be any constant complex number then lim?(z?z_0 )??c=c ?
lim?(z?z_0 )??z^n ?=?z_0?^n
lim?(z?z_0 )??z ?^n ?=(z_0 ) ?^n (or lim?(z?z_0 )??z^n ?=?z_0?^n)
theorem:-
suppose that
lim?(z?z_0 )??f(z)=L? , lim?(z?z_0 )??g(z)=m ,then ?
lim?(z?z_0 )??[f(z)±g(z)]=L±m?
lim?(z?z_0 )??[f(z).g(z)]=L.m?
lim?(z?z_0 )??[f(z)/g(z) ]=L/m? if g(z)?0