Consider the generalized discounted Hamilton-Jacobi equation
\[
λ
a
(
x
)
u
+
H
(
x
,
D
u
)
=
c
(
H
)
,
\lambda a(x)u+H(x,Du)=c(H),
\]
where
a
(
x
)
a(x)
may vanish or change the signs. Two examples are given in this paper showing that the viscosity solutions of the above equation may not converge as
λ
\lambda
tends to zero.