\[ \begin{split} &\lim_{n \to \infty} \frac{(n+2)!+(n+1)!}{(n+2)!-(n+1)!}=\\[16pt] &=\lim_{n \to \infty} \frac{(n+1)!\cdot (n+2)+(n+1)!}{(n+1)!\cdot (n+2)-(n+1)!}=\\[16pt] &=\lim_{n \to \infty} \frac{(n+1)!\cdot (n+2+1)}{(n+1)!\cdot (n+2-1)}=\\[16pt] &=\lim_{n \to \infty} \frac{(n+1)!\cdot (n+3)}{(n+1)!\cdot (n+1)}=\\[16pt] &=\lim_{n \to \infty} \frac{n+3}{n+1}\cdot \frac{\frac{1}{n}}{\frac{1}{n}}=\\[16pt] &=\lim_{n \to \infty} \frac{1+\frac{3}{n}}{1+\frac{1}{n}}=\\[16pt] &=\frac{1}{1}=1 \end{split} \]