Question

The value of \(\underset{n\rightarrow \infty }{lim}\frac{(n+2)!+(n+1)!}{(n+2)!-(n+1)!}\) is ____

• A. \(-1\)
• B. \(0\)
• C. \(1\)
• D. \(2\)

Hint:

Take \((n+1)!\) common from top and bottom.

Answer 1

The Correct Option is C

Step 1: Plan:
Divide top and bottom by $(n+1)!$ at once.

Step 2: Divide:
$\frac{(n+2)!}{(n+1)!} = n+2$ and $\frac{(n+1)!}{(n+1)!} = 1$. So the expression is $\frac{(n+2)+1}{(n+2)-1} = \frac{n+3}{n+1}$.
For large $n$, the extra $2$ hardly matters compared with $n$, and the fraction goes to $1$.

Final Answer:
The limit is $1$, option (C). \[ \boxed{1} \]