calculus - Limit of Exponential Integral Function

I want to ask how to prove whether the following limit is corrent
$$\lim_{x \to 0} \left[ {x{e^x}{E_1}( x )} \right] = 0,$$
with

$\displaystyle {E_1}( x ) = \int_x^\infty \frac{e^{ - t}}{t}dt$. I try to run it by Matlab and it seems to be true.