Question

$\underset{n\to \infty }{\lim }\left[\frac{\left(n+2\right)!+\left(n+1\right)!}{\left(n+2\right)!-\left(n+1\right)!}\right]$

Answer 1

$$\begin{gathered} \underset{n\to \infty }{\lim }\left[\frac{\left(n+2\right)!+\left(n+1\right)!}{\left(n+2\right)!-\left(n+1\right)!}\right] \\ =\underset{n\to \infty }{\lim }\left[\frac{\left(n+2\right)\left(n+1\right)!+\left(n+1\right)!}{\left(n+2\right)\left(n+1\right)!-\left(n+1\right)!}\right] \\ =\underset{n\to \infty }{\lim }\left[\frac{\left(n+1\right)!}{\left(n+1\right)!}\times \frac{\left(n+2+1\right)}{\left(n+2-1\right)}\right] \\ =\underset{n\to \infty }{\lim }\left[\frac{n+3}{n+1}\right] \end{gathered}$$

Dividing the numerator and the denominator by n:

$$\begin{gathered} \underset{n\to \infty }{\lim }\left[\frac{1+{\displaystyle \frac{3}{n}}}{1+{\displaystyle \frac{1}{n}}}\right] \\ \mathrm{When}n\to \infty ,\text{then}\frac{1}{n}\to 0. \\ \Rightarrow \frac{1}{1}=1 \end{gathered}$$