limits involving the point at infinity

limits involving the point at infinity?-
for the statement
lim?(z?z_0 )??f(z)=w_0 ?
when either z_0 or w_0 ,or possibly each of these numbers ,is replaced by the
point at infinity we simply replace the appropriate neighborhoods
of z_0 and w_0 by neighborhoods of ?.
the statment
lim?(z?z_0 )??f(z)=?.?
for instance ,means that ? ?>0 ,? ?>0 ? |f(z)|>1/? whenever
0<|z-z_0 |that is ,the point w=f(z)lies in the ?-neighborhood
|w|>1/? of ? whenever z lies in the deleted neighborhood
0<|z-z_0 ||1/f(z) -0|we see that
lim?(z?z_0 )??f(z)=? ? iff lim?(z?z_0 )??1/f(z) =0…(2)?
to say that
lim?(z??)??f(z)=w_0 ?
means that ,? ? >0 ,? ?>0 ?
|f(z)-w_0 |1/?…(3)
finally ,the statement
lim?(z??)??f(z)=??
is to be interpreted as saying that ,??>0 ? ?>0
|f(z)|>1/? whenever |z|>1/?…(5)
when z is replaced by 1/z,this statement can be put in the form
|1/f(1/z) -0|and so
lim?(z??)??f(z)=? iff ? lim?(z?0)???1/f(1/z) =0…(6)? ?

replacing z by 1/z in statement (3)and then writing the result as
|f(1/z)-w_0 |we find that
lim?(z??)??f(z)=w_(0 ) ? iff lim?(z?0)??f(1/z)=w_0 ?…(4)