`lim_(n to oo)(n!)/((n+1)!-n!)`

Options

A. `0`
B. `2`
C. `-1`
D. `-2`

Answer

To solve the limit \( \lim_{n \to \infty} \frac{n!}{(n+1)! - n!} \), we can follow these steps:

### Step 1: Rewrite the factorials
We know that \( (n+1)! = (n+1) \cdot n! \). Thus, we can rewrite the expression in the limit:

\[ (n+1)! - n! = (n+1) \cdot n! - n! = n! \cdot (n + 1 - 1) = n! \cdot n \]

### Step 2: Substitute back into the limit
Now, substituting this back into the limit gives us:

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

### Step 3: Simplify the expression
We can simplify the expression by canceling \( n! \) in the numerator and the denominator:

\[ \lim_{n \to \infty} \frac{1}{n} \]

### Step 4: Evaluate the limit
Now we can evaluate the limit as \( n \) approaches infinity:

\[ \lim_{n \to \infty} \frac{1}{n} = 0 \]

### Final Answer
Thus, the final answer is:

\[ \lim_{n \to \infty} \frac{n!}{(n+1)! - n!} = 0 \]

---