`lim_(nrarroo)((r)^(1//n))/n` equals

Options

A. e
B. `e^(-1)`
C. 1
D. None

Answer

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

### Step 1: Rewrite the limit expression
We start with the limit:
\[ \lim_{n \to \infty} \frac{r^{1/n}}{n} \]

### Step 2: Analyze \( r^{1/n} \)
As \( n \) approaches infinity, we know that \( r^{1/n} \) approaches \( 1 \). This is because:
\[ r^{1/n} = e^{\frac{\ln(r)}{n}} \quad \text{and as } n \to \infty, \frac{\ln(r)}{n} \to 0 \]
Thus, \( r^{1/n} \to e^0 = 1 \).

### Step 3: Substitute the limit
Now substituting this back into our limit expression, we have:
\[ \lim_{n \to \infty} \frac{1}{n} \]

### Step 4: Evaluate the limit
As \( n \) approaches infinity, \( \frac{1}{n} \) approaches \( 0 \). Therefore:
\[ \lim_{n \to \infty} \frac{1}{n} = 0 \]

### Conclusion
Thus, the final result is:
\[ \lim_{n \to \infty} \frac{r^{1/n}}{n} = 0 \]

### Summary of Steps
1. Rewrite the limit expression.
2. Analyze \( r^{1/n} \) as \( n \to \infty \).
3. Substitute the limit into the expression.
4. Evaluate the limit to find the final answer.