# Limit to Infinity of Exponential Integral Function

## Theorem

$\displaystyle \lim_{x \mathop \to \infty} \map \Ei x = 0$

where $\Ei$ denotes the exponential integral function.

## Proof

 $\displaystyle \lim_{x \mathop \to \infty} \map \Ei x$ $=$ $\displaystyle \lim_{x \mathop \to \infty} \int_x^\infty \frac {e^{-u} } u \rd u$ Definition of Exponential Integral Function $\displaystyle$ $=$ $\displaystyle \lim_{x \mathop \to \infty} \int_1^\infty \frac {e^{-x t} } {x t} x \rd t$ substituting $u = x t$ $\displaystyle$ $=$ $\displaystyle \int_1^\infty \lim_{x \mathop \to \infty} \paren {\frac {e^{-x t} } t} \rd t$ Lebesgue's Dominated Convergence Theorem $\displaystyle$ $=$ $\displaystyle \int_1^\infty \frac 0 t \rd t$ Exponential Tends to Zero and Infinity $\displaystyle$ $=$ $\displaystyle 0$

$\blacksquare$