This a pretty weird question... because:

$\displaystyle E(e^{tX}) = M(t) = \int_0^{\infty} e^{xt} e^{-x} dx = \int_0^{\infty} e^{-x(1-t)}dx = \lim_{k \to \infty} \left[\frac{e^{x(t-1)}}{t-1}\right]_0^k$

But the limit: $\displaystyle \lim_{k \to \infty} \left[\frac{e^{k(t-1)}}{t-1}\right]$ is undefined?

How am I meant to compute the MGF then?

Thanks