Question

The value of $\underset{x\to \infty }{\lim }\frac{n!}{\left(n+1\right)!-n!}$ is

(a) 0

(b) 1

(c) −1

(d) none of these

Answer 1

(a) 0
$$\begin{gathered} \underset{n\to \infty }{\lim }\frac{n!}{\left(n+1\right)!-n!} \\ \mathrm{Divi}\text{ding}{\mathrm{N}}^r\mathrm{and}{\mathrm{D}}^r\mathrm{by}n!: \\ \underset{n\to \infty }{\lim }\frac{1}{{\displaystyle \frac{\left(n+1\right)!}{n!}}-1} \\ =\underset{n\to \infty }{\lim }\frac{1}{{\displaystyle \frac{\left(n+1\right)n!}{n!}}-1} \\ =\underset{n\to \infty }{\lim }\frac{1}{n+1-1} \\ =\underset{n\to \infty }{\lim }=\frac{1}{n}=0 \end{gathered}$$