Kummer's Hypergeometric Theorem/Lemma 1

Lemma for Kummer's Hypergeometric Theorem

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

where $y^{\underline k}$ denotes the $k$th falling factorial of $y$.

Proof
From Properties of Limit at Infinity of Real Function: Product Rule, we have:

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

In the present example, for the $k$th limit, we have:
 * $\map {f_k} y = \paren {y + 1 - k}$
 * $\map {g_k} y = \paren {y + n + k}$

Therefore taking the derivative of the numerator $\map {f_k} y$ and denominator $\map {g_k} y$ $y$, we proceed:

Therefore: