How Does Asymptotic Expansion Apply to Integrals with Oscillatory Behavior?

[Clausius2]

Hey guys,
I need help with the expansion of this integral:

\int_0^\infty Z(x) J_o(\lambda x)dx for \lambda \rightarrow \infty

where I know that Z(x)\sim x^\sqrt{2} for small x and
exponentially small for large x

It seems with other examples that I have done that the major contribution to the integral comes from the region x\sim 1/\lambda. For larger x the integrand oscillates rapidly and the integration cancels. One change of variable (re-scaling) that you may try is t=\lambda x. But if you do it you end up with a divergent integral. And at first glance the original integral is convergent. Any hints?

Thanks.

 

[Clausius2]

with divergent integral I mean that I arrive to this integral after the rescalement:

\frac{1}{\lambda^{\sqrt{2}+1}}\int_0^\infty t^{\sqrt{2}} J_o(t)dt