`lim_(nrarr oo) ((nsqrt(a)+nsqrt(b))/(2))^n,a,b,gt 0` equals

Options

A. `1`
B. `sqrt(ab)`
C. `ab`
D. `(2b)/(2)`

Explanation

To solve the limit \( \lim_{n \to \infty} \left( \frac{n \sqrt{a} + n \sqrt{b}}{2} \right)^n \) where \( a, b > 0 \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the limit:
\[ L = \lim_{n \to \infty} \left( \frac{n \sqrt{a} + n \sqrt{b}}{2} \right)^n \]
We can factor out \( n \) from the numerator:
\[ L = \lim_{n \to \infty} \left( n \left( \frac{\sqrt{a} + \sqrt{b}}{2} \right) \right)^n \]

### Step 2: Simplify the expression
This can be rewritten as:
\[ L = \lim_{n \to \infty} n^n \left( \frac{\sqrt{a} + \sqrt{b}}{2} \right)^n \]

### Step 3: Analyze the limit
As \( n \to \infty \), the term \( n^n \) grows very rapidly. The term \( \left( \frac{\sqrt{a} + \sqrt{b}}{2} \right)^n \) will also grow, but we need to determine the behavior of the entire expression.

### Step 4: Use logarithmic properties
To analyze the limit more effectively, we can take the natural logarithm:
\[ \ln L = \lim_{n \to \infty} \left( n \ln n + n \ln \left( \frac{\sqrt{a} + \sqrt{b}}{2} \right) \right) \]
This can be split into two parts:
\[ \ln L = \lim_{n \to \infty} n \ln n + n \ln \left( \frac{\sqrt{a} + \sqrt{b}}{2} \right) \]

### Step 5: Evaluate the limit
As \( n \to \infty \), the term \( n \ln n \) dominates the expression. Therefore, we can conclude:
\[ \ln L \to \infty \quad \text{(since \( n \ln n \) grows faster than any constant)} \]
Thus, \( L \to \infty \).

### Conclusion
The limit diverges to infinity:
\[ \lim_{n \to \infty} \left( \frac{n \sqrt{a} + n \sqrt{b}}{2} \right)^n = \infty \]