`lim_(nto oo) ((nsqrt(p)+nsqrt(q))/(2))^n,p,q,gt 0` equals

Options

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

Explanation

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

### Step 1: Rewrite the expression
We start with the limit:
\[ L = \lim_{n \to \infty} \left( \frac{n\sqrt{p} + n\sqrt{q}}{2} \right)^n \]
This can be simplified to:
\[ L = \lim_{n \to \infty} \left( \frac{n(\sqrt{p} + \sqrt{q})}{2} \right)^n \]

### Step 2: Factor out \( n \)
Now, we can factor out \( n \):
\[ L = \lim_{n \to \infty} \left( n \cdot \frac{\sqrt{p} + \sqrt{q}}{2} \right)^n \]

### Step 3: Analyze the limit
As \( n \to \infty \), \( n \cdot \frac{\sqrt{p} + \sqrt{q}}{2} \) approaches infinity since \( \frac{\sqrt{p} + \sqrt{q}}{2} > 0 \). Therefore, we can express this as:
\[ L = \lim_{n \to \infty} n^n \left( \frac{\sqrt{p} + \sqrt{q}}{2} \right)^n \]

### Step 4: Use the exponential form
To analyze this limit, we can rewrite it in exponential form:
\[ L = \lim_{n \to \infty} e^{n \ln \left( n \cdot \frac{\sqrt{p} + \sqrt{q}}{2} \right)} \]

### Step 5: Simplify the logarithm
Now we simplify the logarithm:
\[ \ln \left( n \cdot \frac{\sqrt{p} + \sqrt{q}}{2} \right) = \ln n + \ln \left( \frac{\sqrt{p} + \sqrt{q}}{2} \right) \]
Thus,
\[ L = \lim_{n \to \infty} e^{n \left( \ln n + \ln \left( \frac{\sqrt{p} + \sqrt{q}}{2} \right) \right)} \]

### Step 6: Analyze the limit of the exponent
The term \( n \ln n \) grows much faster than \( n \ln \left( \frac{\sqrt{p} + \sqrt{q}}{2} \right) \) as \( n \to \infty \). Therefore, we can conclude:
\[ L \to \infty \]

### Final Result
Thus, the limit diverges to infinity:
\[ L = \infty \]