`lim_(n to oo) x^(2)sin(log_(e)sqrt(cos((pi)/(x))))` is less then

Options

A. `0`
B. `-(pi^(2))/(2)`
C. `-(pi^(2))/(4)`
D. `(pi^(2))/(8)`

Explanation

To solve the limit \( \lim_{x \to \infty} x^2 \sin\left(\log_e\sqrt{\cos\left(\frac{\pi}{x}\right)}\right) \), we will follow these steps:

### Step 1: Simplify the expression
We start with the limit:
\[ \lim_{x \to \infty} x^2 \sin\left(\log_e\sqrt{\cos\left(\frac{\pi}{x}\right)}\right) \]
We can rewrite the sine function using the logarithmic identity:
\[ \sin(a) \approx a \quad \text{for small } a \]
Thus, we can approximate:
\[ \sin\left(\log_e\sqrt{\cos\left(\frac{\pi}{x}\right)}\right) \approx \log_e\sqrt{\cos\left(\frac{\pi}{x}\right)} \]

### Step 2: Rewrite the logarithm
Using the property of logarithms:
\[ \log_e\sqrt{y} = \frac{1}{2} \log_e y \]
we can express our limit as:
\[ \lim_{x \to \infty} x^2 \cdot \frac{1}{2} \log_e\left(\cos\left(\frac{\pi}{x}\right)\right) \]

### Step 3: Analyze \(\cos\left(\frac{\pi}{x}\right)\)
As \(x \to \infty\), \(\frac{\pi}{x} \to 0\). Therefore, we can use the Taylor expansion for \(\cos\):
\[ \cos\left(\frac{\pi}{x}\right) \approx 1 - \frac{1}{2}\left(\frac{\pi}{x}\right)^2 \]
Thus, we have:
\[ \log_e\left(\cos\left(\frac{\pi}{x}\right)\right) \approx \log_e\left(1 - \frac{1}{2}\left(\frac{\pi}{x}\right)^2\right) \]

### Step 4: Use the logarithmic approximation
Using the approximation \(\log_e(1 + u) \approx u\) for small \(u\):
\[ \log_e\left(1 - \frac{1}{2}\left(\frac{\pi}{x}\right)^2\right) \approx -\frac{1}{2}\left(\frac{\pi}{x}\right)^2 \]
So, we can substitute this back into our limit:
\[ \lim_{x \to \infty} x^2 \cdot \frac{1}{2} \left(-\frac{1}{2}\left(\frac{\pi}{x}\right)^2\right) \]

### Step 5: Simplify the limit
This simplifies to:
\[ \lim_{x \to \infty} x^2 \cdot \frac{-\pi^2}{4x^2} = \lim_{x \to \infty} \frac{-\pi^2}{4} = -\frac{\pi^2}{4} \]

### Final Result
Thus, the limit is:
\[ \lim_{x \to \infty} x^2 \sin\left(\log_e\sqrt{\cos\left(\frac{\pi}{x}\right)}\right) = -\frac{\pi^2}{4} \]

### Conclusion
The answer is:
\[ \text{Option 3: } -\frac{\pi^2}{4} \]