Question

Write the value of $\underset{n\to \infty }{\lim }\frac{n!+\left(n+1\right)!}{\left(n+1\right)!+\left(n+2\right)!}.$

Answer 1

$$\begin{gathered} \underset{n\to \infty }{\lim }\left[\frac{n!+\left(n+1\right)!}{\left(n+1\right)!+\left(n+2\right)!}\right] \\ =\underset{n\to \infty }{\lim }\left[\frac{n!+\left(n+1\right)n!}{\left(n+1\right)!+\left(n+2\right)\left(n+1\right)!}\right] \\ =\underset{n\to \infty }{\lim }\left[\frac{n!\left(1+n+1\right)}{\left(n+1\right)!\left(1+n+2\right)}\right] \\ =\underset{n\to \infty }{\lim }\left[\frac{n!\left(n+2\right)}{\left(n+1\right)n!\left(n+3\right)}\right] \\ =\underset{n\to \infty }{\lim }\left[\frac{n+2}{\left(n+1\right)\left(n+3\right)}\right] \\ \mathrm{D}\text{egree of the denominator is greater than that of the numerator.} \\ \Rightarrow 0 \end{gathered}$$