Asymptotic expansion integral

Clausius2

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

[tex]\int_0^\infty Z(x) J_o(\lambda x)dx[/tex] for [tex]\lambda \rightarrow \infty[/tex]

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

It seems with other examples that I have done that the major contribution to the integral comes from the region [tex]x\sim 1/\lambda[/tex]. For larger [tex]x[/tex] the integrand oscillates rapidly and the integration cancels. One change of variable (re-scaling) that you may try is [tex]t=\lambda x[/tex]. 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:

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