Can asymptotic of a Mellin (or laplace inverse ) be evaluated? I mean, given the Mellin inverse integral $ \int_{c-i\infty}^{c+i\infty}dsF(s)x^{-s} $, can we evaluate this integral, at least as $ x \rightarrow \infty $?
Can the same be made for $ \int_{c-i\infty}^{c+i\infty}dsF(s)\exp(st) $ as $ x \rightarrow \infty $?
Why or why not can this be evaluated in order to get the asymptotic behaviour of Mellin inverse transforms?

yes we can evaluate above integral but it depends on F(s).what is your F(s).then we can see how to solve it.above integral is inverse mellin transform.Some times it is v difficult to find inverse,it all depends on what F(s) is.
