$| \Gamma (x_{n})| \to 0 $ as $n \to \infty$ where $x_{n} \in [-n+\delta,1-n-\delta] $

Question

I'm trying to prove $| \Gamma (x_{n})| \to 0 $ as $n \to \infty$ where $x_{n} \in [-n+\delta,1-n-\delta] $ for some $\delta > 0 $.

What possible method can I use?

As a special case, I've proved the above result for $x_{n} = 1/2 - n$. Can I use this result for the proof of the above?

Answer 1

From $\Gamma(x+1)=x\,\Gamma(x)$ you get $$\Gamma(-n+x)=\frac{1}{(x-n)(x-(n-1))\dots(x-1)}\Gamma(x).$$ If $x_n=-n+a_n$, so that $\delta<a_n<1-\delta$, you have $\Gamma(a_n)$ bounded and $$\frac{1}{(a_n-n)(a_n-(n-1))\dots(a_n-1)}\to 0.$$

Answer 2

This is false. The gamma function has poles at the negative integers, so you can choose arbitrarily large moduli for each $n$.