`lim_(ntooo) ((n!)^(1//n))/(n)` equals

Options

A. e
B. `e^(-1)`
C. 1
D. none of these

Explanation

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

### Step 1: Rewrite the Limit
We start by rewriting the limit:
\[ a = \lim_{n \to \infty} \frac{(n!)^{1/n}}{n} \]
This can be rearranged as:
\[ a = \lim_{n \to \infty} \frac{n!}{n^n}^{1/n} \]

### Step 2: Take the Natural Logarithm
Taking the natural logarithm of both sides gives:
\[ \log a = \lim_{n \to \infty} \left( \log(n!) - n \log n \right) \cdot \frac{1}{n} \]

### Step 3: Use Stirling's Approximation
Using Stirling's approximation, we know that:
\[ n! \sim \sqrt{2 \pi n} \left( \frac{n}{e} \right)^n \]
Thus, we can write:
\[ \log(n!) \sim \frac{1}{2} \log(2 \pi n) + n \log n - n \]

### Step 4: Substitute Stirling's Approximation
Substituting this into our expression for \(\log a\):
\[ \log a = \lim_{n \to \infty} \left( \frac{1}{2} \log(2 \pi n) + n \log n - n - n \log n \right) \cdot \frac{1}{n} \]
This simplifies to:
\[ \log a = \lim_{n \to \infty} \left( \frac{1}{2n} \log(2 \pi n) - 1 \right) \]

### Step 5: Evaluate the Limit
As \( n \to \infty \), the term \(\frac{1}{2n} \log(2 \pi n)\) approaches \(0\). Therefore:
\[ \log a = -1 \]

### Step 6: Solve for \(a\)
Exponentiating both sides gives:
\[ a = e^{-1} = \frac{1}{e} \]

### Conclusion
Thus, the limit is:
\[ \lim_{n \to \infty} \frac{(n!)^{1/n}}{n} = \frac{1}{e} \]

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