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: Understanding the Question:
We write \((n+2)! = (n+2)(n+1)!\) so that every term has a common factor.

Step 2: Simplify:
Numerator: \((n+1)!\big[(n+2) + 1\big] = (n+1)!\,(n+3)\).
Denominator: \((n+1)!\big[(n+2) - 1\big] = (n+1)!\,(n+1)\).
The ratio is \(\dfrac{n+3}{n+1}\).

Step 3: Take the limit:
\[ \lim_{n\to\infty}\frac{n+3}{n+1} = \lim_{n\to\infty}\frac{1+3/n}{1+1/n} = 1 \]

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