Kummer's Hypergeometric Theorem/Lemma 2

Lemma for Kummer's Hypergeometric Theorem

 * $\ds \lim_{y \mathop \to \infty} \dfrac {\paren {y + \dfrac n 2 + 1}^{\overline x} } {\paren {y + n + 1}^{\overline x} } = 1$

where $y^{\overline x}$ denotes the $x$th rising factorial of $y$.

Proof
From L'Hôpital's Rule:Corollary 2, we have:
 * $\ds \lim_{x \mathop \to a^+} \frac {\map f x} {\map g x} = \lim_{x \mathop \to a^+} \frac {\map {f'} x} {\map {g'} x}$

Therefore taking the the derivative of the numerator and denominator $y$, we obtain:
 * $\ds \lim_{y \mathop \to \infty}\paren {\dfrac {\paren {y + \dfrac n 2 + x} } {\paren {y + n + x} } } =\dfrac 1 1 = 1$