Evaluate `lim_(nto oo)((n+2)!+(n+1)!)/((n+2)!-(n+1)!)`

Answer

To evaluate the limit

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

we will follow these steps:

### Step 1: Rewrite the factorials
We can express the factorials in terms of \( (n+1)! \):

\[ (n+2)! = (n+2)(n+1)!. \]

So, we can rewrite the limit as:

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

### Step 2: Factor out \( (n+1)! \)
Now, we can factor \( (n+1)! \) from both the numerator and the denominator:

\[ = \lim_{n \to \infty} \frac{(n+1)! \left( (n+2) + 1 \right)}{(n+1)! \left( (n+2) - 1 \right)}. \]

### Step 3: Simplify the expression
The \( (n+1)! \) cancels out:

\[ = \lim_{n \to \infty} \frac{n + 3}{n + 1}. \]

### Step 4: Divide by \( n \)
To simplify further, we can divide both the numerator and the denominator by \( n \):

\[ = \lim_{n \to \infty} \frac{1 + \frac{3}{n}}{1 + \frac{1}{n}}. \]

### Step 5: Evaluate the limit
Now, as \( n \to \infty \), \( \frac{3}{n} \) and \( \frac{1}{n} \) both approach 0:

\[ = \frac{1 + 0}{1 + 0} = \frac{1}{1} = 1. \]

### Final Answer
Thus, the limit is

\[ \boxed{1}. \]