Limit to Infinity of x minus Gamma of Reciprocal of x

Theorem

$\ds \lim_{x \mathop \to \infty} \paren {x - \map \Gamma {\dfrac 1 x} } = \gamma$

where:

$\Gamma$ denotes the $\Gamma$ (Gamma) function
$\gamma$ denotes the Euler-Mascheroni constant.

Proof

 $\ds \lim_{x \mathop \to \infty} \paren {x - \map \Gamma {\frac 1 x} }$ $=$ $\ds \lim_{x \mathop \to 0} \paren {\frac 1 x - \map \Gamma x}$ $\ds$ $=$ $\ds \lim_{x \mathop \to 0} \paren {\frac 1 x - \frac {\map \Gamma {x + 1} } x}$ Gamma Difference Equation $\ds$ $=$ $\ds \lim_{x \mathop \to 0} \paren {\frac {\map \Gamma 1 - \map \Gamma {1 + x} } x}$ $\map \Gamma 1 = 1$ as Gamma Function Extends Factorial $\ds$ $=$ $\ds -\map {\Gamma'} 1$ Definition of Derivative $\ds$ $=$ $\ds \gamma$ Derivative of Gamma Function at 1

$\blacksquare$

Historical Note

François Le Lionnais and Jean Brette, in their Les Nombres Remarquables of $1983$, attribute this to a $1976$ result of K. Demys, but this has not been corroborated.