= Solution
The complete integral is $\Gamma(1/2)=\sqrt\pi$. Therefore apply <asymptotic expansion by repeated integration by parts> to the tail $I(x)=\int_x^\infty t^{-1/2}e^{-t}\,dt$:
$$
I(x)=e^{-x}x^{-1/2}-\frac12\int_x^\infty t^{-3/2}e^{-t}\,dt
=e^{-x}\left(x^{-1/2}-\frac12x^{-3/2}+\frac34x^{-5/2}\right)-\frac{15}8\int_x^\infty t^{-7/2}e^{-t}\,dt.
$$
The last integral is bounded by $x^{-7/2}e^{-x}$. Thus
$$
\boxed{f(x)=\sqrt\pi-e^{-x}x^{-1/2}+\tfrac12e^{-x}x^{-3/2}-\tfrac34e^{-x}x^{-5/2}+O(e^{-x}x^{-7/2}).}
$$
In particular the first three additive terms are $\sqrt\pi$, $-e^{-x}x^{-1/2}$ and $\tfrac12e^{-x}x^{-3/2}$; the formula also supplies three successive terms of the exponentially small tail if the constant is counted separately. A power expansion alone would retain only $\sqrt\pi$ and miss this information.
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