Mathematics - Limits Question with Solution | TestHub

${\mathrm{Lt}}_{n\to \infty }\left(\frac{a_1+1}{a_1}\right)\left(\frac{a_2+1}{a_2}\right)\cdots \cdots \left(\frac{a_n+1}{a_n}\right)$

$$\begin{array}{rl}& {\mathrm{Lt}}_{n\to \infty }\left(\frac{a_2}{2}\right)\left(\frac{a_3}{3}\right)\left(\frac{a_4}{4}\right)\dots \left(\frac{a_{n+1}}{n+1}\right)\frac{1}{a_1{a}_2\dots a_n}\\ & ={\mathrm{Lt}}_{n\to \infty }\frac{a_{n+1}}{(n+1)!}={\mathrm{Lt}}_{n\to \infty }\frac{1+a_n}{n!}={\mathrm{Lt}}_{n\to \infty }\left(\frac{1}{n!}+\frac{a_n}{n!}\right)\\ & ={\mathrm{Lt}}_{n\to \infty }\left(\frac{1}{n!}+\frac{1}{(n-1)!}+\frac{a_{n-1}}{(n-1)!}\right)=e\end{array}$$