Calculus

Divide both sides by $(n+2)!$ .

$= \lim_{n \rightarrow \infty} \dfrac{ 1 + \frac{1}{n+2}}{1 - \frac{1}{n+2}}$

$n \rightarrow \infty$ implies $n+2 \rightarrow \infty$ implies $\frac{1}{n+2} \rightarrow 0$

Therefore,

$\lim_{n \rightarrow \infty} \dfrac{ 1 + \frac{1}{n+2}}{1 - \frac{1}{n+2}} = \dfrac{1 + 0}{1-0} = \boxed{1}$