We consider the family of rational maps
F
λ
(
z
)
=
z
n
+
λ
/
z
d
F_\lambda (z) = z^n + \lambda /z^d
, where
n
,
d
≥
2
n,d \geq 2
and
λ
\lambda
is small. If
λ
\lambda
is equal to 0, the limiting map is
F
0
(
z
)
=
z
n
F_0(z)=z^n
and the Julia set is the unit circle. We investigate the behavior of the Julia sets of
F
λ
F_\lambda
when
λ
\lambda
tends to 0, obtaining two very different cases depending on
n
n
and
d
d
. The first case occurs when
n
=
d
=
2
n=d=2
; here the Julia sets of
F
λ
F_\lambda
converge as sets to the closed unit disk. In the second case, when one of
n
n
or
d
d
is larger than
2
2
, there is always an annulus of some fixed size in the complement of the Julia set, no matter how small
|
λ
|
|\lambda |
is.