  It is known that local zeta functions associated with real analytic functions
can be analytically continued as meromorphic functions to the hole complex
plane. In this paper, certain cases of specific (non-real analytic) smooth
functions are precisely investigated. In particular, we give asymptotic limits
of local zeta functions at some singularities along one direction. It follows
from the behaviors that these local zeta functions have singularities different
from poles. Then we show the optimality of the lower estimates of a certain
quantity concerning with meromorphic continuation of local zeta functions in
the case of all smooth functions expressed as $u(x,y)x^a y^b +$ flat function,
where $u(0,0)\neq 0$ and $a,b$ are nonnegative integers satisfying $a\neq b$.
