The value of $\lim_{n\to\infty} \dfrac{(n+2)! + (n+1)!}{(n+2)! - (n+1)!}$ is ____

The given limit is $\lim_{n\to\infty} \dfrac{(n+2)! + (n+1)!}{(n+2)! - (n+1)!}$

Taking $(n+1)!$ common from both numerator and denominator:

$\lim_{n\to\infty} \dfrac{(n+1)![(n+2) + 1]}{(n+1)![(n+2) - 1]}$

Cancelling $(n+1)!$ from numerator and denominator:

$\lim_{n\to\infty} \dfrac{n+3}{n+1}$

Dividing numerator and denominator by $n$:

$\lim_{n\to\infty} \dfrac{1 + \dfrac{3}{n}}{1 + \dfrac{1}{n}}$

As $n \to \infty$, $\dfrac{1}{n} \to 0$. Substituting the limit:

$\dfrac{1 + 0}{1 + 0} = 1$

Answer: $1$