Ответ:

Распишем факториалы .  

[tex]\bf \lim\limits_{n \to +\infty}\dfrac{(n+2)!-(n+1)!}{(n+1)!+n!}=\lim\limits_{n \to +\infty}\dfrac{(n+1)!\ (n+2)\ -\ (n+1)!}{n!\ (n+1)\ +\ n!}=\\\\\\=\lim\limits_{n \to +\infty}\dfrac{(n+1)!\cdot (n+2-1)}{n!\cdot (n+1+1)}=\lim\limits_{n \to +\infty}\dfrac{n!\ (n+1)\cdot (n+1)}{n!\cdot (n+2)}=[/tex]  

[tex]\bf =\lim\limits_{n \to +\infty}\dfrac{(n+1)^2}{n+2}=\lim\limits_{n \to +\infty}\dfrac{n^2+2n+1}{n+2}=\lim\limits_{n \to +\infty}\dfrac{1+\dfrac{2}{n}+\dfrac{1}{n^2}}{\dfrac{1}{n}+\dfrac{2}{n^2}}=\\\\\\=\Big[\dfrac{1}{0}\Big]=\infty[/tex]