integration - Laplace transform of $frac{1}{t}$ and resulting improper integral

Could someone please explain how

$$\int_0^{\infty}\frac{e^{-x}}{x}dx$$ diverges?
This is because the Laplace transform of $\frac{1}{t}$ can be reduced to this integral which has to diverge. But the limit comparison test with $e^{-x}$ shows that the integral converges.

Please help.

Thanks in advance.

Answer

Regarding the second part of the question, to add to carmichael561's answer,

To use the limit comparison test with $e^{-x}$, we need

$$\frac{e^{-x}}{x}\le e^{-x}$$
throughout the interval over which integral takes place.

However, this is true only for $x\gt1$, and hence the test cannot be used.