Question

$\underset{n\to \infty }{lim}\frac{{}^n{P}_n}{{}^{n+1}{P}_{n+1}{-}^n{P}_n}=$

Answer 1

${lim}_{x\to \infty }\frac{{}^n{P}_n}{{}^{n+1}{P}_{n+1}{-}^n{P}_n}$

Since ${}^n{P}_n=n!$

$={lim}_{x\to \infty }\frac{n!}{\left(n+1\right)!-n!}$

$={lim}_{x\to \infty }\frac{1}{\left(1+\left(1/n\right)\right)!-1!}$

$={lim}_{x\to \infty }\frac{1}{n}=0$