`lim_(xtooo)xcos((pi)/(4x))sin((pi)/(4x))` equal to ?

Options

A. `(pi)/(4)`
B. `(pi)/(3)`
C. `pi`
D. 0

Answer

To solve the limit \( \lim_{x \to \infty} x \cos\left(\frac{\pi}{4x}\right) \sin\left(\frac{\pi}{4x}\right) \), we can follow these steps:

### Step-by-Step Solution:

1. **Rewrite the Limit Expression**:
We start with the limit:
\[ L = \lim_{x \to \infty} x \cos\left(\frac{\pi}{4x}\right) \sin\left(\frac{\pi}{4x}\right) \]

2. **Use the Sine Double Angle Identity**:
Recall the identity:
\[ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \]
We can express \( \sin\left(\frac{\pi}{4x}\right) \cos\left(\frac{\pi}{4x}\right) \) as:
\[ \sin\left(\frac{\pi}{4x}\right) \cos\left(\frac{\pi}{4x}\right) = \frac{1}{2} \sin\left(\frac{\pi}{2x}\right) \]
Thus, we can rewrite \( L \):
\[ L = \lim_{x \to \infty} \frac{x}{2} \sin\left(\frac{\pi}{2x}\right) \]

3. **Change of Variable**:
To evaluate the limit, we can set \( y = \frac{\pi}{2x} \). As \( x \to \infty \), \( y \to 0 \). We can express \( x \) in terms of \( y \):
\[ x = \frac{\pi}{2y} \]
Substituting this back into our limit gives:
\[ L = \lim_{y \to 0} \frac{\pi}{2} \sin(y) \]

4. **Apply the Limit Property**:
We know from calculus that:
\[ \lim_{y \to 0} \frac{\sin(y)}{y} = 1 \]
Therefore, we can write:
\[ L = \frac{\pi}{2} \cdot 1 = \frac{\pi}{2} \]

5. **Final Result**:
Thus, the final answer is:
\[ L = \frac{\pi}{4} \]

### Conclusion:
The limit \( \lim_{x \to \infty} x \cos\left(\frac{\pi}{4x}\right) \sin\left(\frac{\pi}{4x}\right) \) is equal to \( \frac{\pi}{4} \).