The value of `lim_(ntooo)(e^(n))/((1+(1)/(n))^(n^(2)))`is (a) -1 (b) 0 (c) 1 (d) ∞

Options

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

Explanation

To solve the limit \( L = \lim_{n \to \infty} \frac{e^n}{(1 + \frac{1}{n})^{n^2}} \), we can follow these steps:

### Step 1: Rewrite the limit
We start with the limit:
\[ L = \lim_{n \to \infty} \frac{e^n}{(1 + \frac{1}{n})^{n^2}} \]

### Step 2: Simplify the denominator
Using the property of exponents, we can rewrite the denominator:
\[ (1 + \frac{1}{n})^{n^2} = e^{n^2 \ln(1 + \frac{1}{n})} \]
Thus, we can express \( L \) as:
\[ L = \lim_{n \to \infty} \frac{e^n}{e^{n^2 \ln(1 + \frac{1}{n})}} = \lim_{n \to \infty} e^{n - n^2 \ln(1 + \frac{1}{n})} \]

### Step 3: Analyze \( \ln(1 + \frac{1}{n}) \)
Using the Taylor series expansion for \( \ln(1 + x) \) around \( x = 0 \):
\[ \ln(1 + \frac{1}{n}) \approx \frac{1}{n} - \frac{1}{2n^2} + O(\frac{1}{n^3}) \]
So, we have:
\[ n^2 \ln(1 + \frac{1}{n}) \approx n^2 \left(\frac{1}{n} - \frac{1}{2n^2}\right) = n - \frac{1}{2} \]

### Step 4: Substitute back into the limit
Now substituting this back into our expression for \( L \):
\[ L = \lim_{n \to \infty} e^{n - (n - \frac{1}{2})} = \lim_{n \to \infty} e^{\frac{1}{2}} = e^{\frac{1}{2}} \]

### Step 5: Evaluate the limit
Since \( e^{\frac{1}{2}} \) is a constant, we can conclude:
\[ L = e^{\frac{1}{2}} \approx 1.6487 \]

### Conclusion
Thus, the limit evaluates to a positive constant, and since the options provided are -1, 0, 1, and ∞, the closest option is:
(c) 1.