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barfing on an integral


Hello All

I'm trying to check my solution to approximations of integrals by the 
method of steepest decent.  The integral i'm approximating is:

   I(x) = int e(x*t - e^t) dt     from t=0 to t=infty

my approximation is:

   J(x) = sqrt(2 Pi/x) * e^{1.5 x log x}

The trouble is that MMA is barfing on I(x).  If I try to do it 
analytically with Integrate[] it gives me a whole bunch of E^Infty 
terms.  Most of which go to zero, but I'm having trouble getting MMA to 
actually evaluate the limit.  N[%] doesn't seem to work, and MMA doesn't 
seem to realize that e^infty / e^e^infty is truly zero.

Then I tried using NIntegrate on I(x), and it gave me a whole bunch of 
underflow errors, and said that the integrand is probably oscillatory 
(which it most certainly is not).

I'm in a very awkward position.  My approximation is only valid for large 
values of x (actually, I think anything over 10 will do).  Yet, MMA barfs 
for large values of x.

Is there some way I can check my approximation??

Much thanks.

Peter

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