Limit to Infinity of Exponential Integral Function

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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
\(\displaystyle \) \(=\) \(\displaystyle 0\)

$\blacksquare$


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