Question-28

Evaluate:

\[ \begin{equation*} \lim\limits _{n\rightarrow \infty } \ \frac{( n+2) !+( n+1) !}{( n+2) !-( n+1) !} \end{equation*} \]

Hint

Take \((n+1)!\) outside.

Answer

\(1\)

Solution

\[ \begin{equation*} \begin{aligned} \lim\limits _{n\rightarrow \infty } \ \frac{( n+2) !+( n+1) !}{( n+2) !-( n+1) !} & =\lim\limits _{n\rightarrow \infty } \ \frac{( n+1) ![( n+2) +1]}{( n+1) ![( n+2) -1]}\\ & \\ & =\lim\limits _{n\rightarrow \infty } \ \frac{n+3}{n+1}\\ & \\ & =\lim\limits _{n\rightarrow \infty } \ \frac{1+\frac{3}{n}}{1+\frac{1}{n}}\\ & \\ & =1 \end{aligned} \end{equation*} \]