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:

$\ds \lim_{x \mathop \to \infty} \paren {x - \map \Gamma {\frac 1 x} }$

$=$

$\ds \lim_{x \mathop \to 0} \paren {\frac 1 x - \map \Gamma x}$

$\ds $

$=$

$\ds \lim_{x \mathop \to 0} \paren {\frac 1 x - \frac {\map \Gamma {x + 1} } x}$

$\ds $

$=$

$\ds \lim_{x \mathop \to 0} \paren {\frac {\map \Gamma 1 - \map \Gamma {1 + x} } x}$

$\ds $

$=$

$\ds -\map {\Gamma'} 1$

$\ds $

$=$

$\ds \gamma$

$\blacksquare$

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.