Question

$\underset{n\to \infty }{\lim }\frac{n!}{\left(n+1\right)!+n!}$ is equal to

(a) $\frac{1}{2}$

(b) 0

(c) 2

(d) 1

Answer 1

(b) 0

$$\begin{gathered} \underset{n\to \infty }{\lim }\left[\frac{n!}{\left(n+1\right)!+n!}\right] \\ =\underset{n\to \infty }{\lim }\left[\frac{n!}{\left(n+1\right)\times n!+n!}\right] \\ =\underset{n\to \infty }{\lim }\left[\frac{n!}{n!\left(n+1+1\right)}\right] \\ =\underset{n\to \infty }{\lim }\left[\frac{1}{n+2}\right] \\ =0 \end{gathered}$$