The value of `underset(x -> (pi)/(2))(lim) ({1 - tan (x/2)}{1-"sin"x})/({1 + tan (X/2)}(pi - 2x)^(3))` equals

Options

A. not exist
B. `1/8`
C. `1/16`
D. `1/32`

Answer

To solve the limit problem given, we will follow a systematic approach. The limit we need to evaluate is:

\[ \lim_{x \to \frac{\pi}{2}} \frac{(1 - \tan(\frac{x}{2}))(1 - \sin x)}{(1 + \tan(\frac{x}{2}))(\pi - 2x)^3} \]

### Step 1: Substitute \( x = \frac{\pi}{2} + h \)

To analyze the limit as \( x \) approaches \( \frac{\pi}{2} \), we substitute \( x = \frac{\pi}{2} + h \), where \( h \to 0 \).

\[ \lim_{h \to 0} \frac{(1 - \tan(\frac{\frac{\pi}{2} + h}{2}))(1 - \sin(\frac{\pi}{2} + h))}{(1 + \tan(\frac{\frac{\pi}{2} + h}{2}))(\pi - 2(\frac{\pi}{2} + h))^3} \]

### Step 2: Simplify the expressions

1. **Evaluate \( \tan(\frac{\frac{\pi}{2} + h}{2}) \)**:
\[ \tan\left(\frac{\frac{\pi}{2} + h}{2}\right) = \tan\left(\frac{\pi}{4} + \frac{h}{2}\right) = \frac{1 + \tan(\frac{h}{2})}{1 - \tan(\frac{h}{2})} \]
As \( h \to 0 \), \( \tan(\frac{h}{2}) \approx \frac{h}{2} \).

2. **Evaluate \( \sin(\frac{\pi}{2} + h) \)**:
\[ \sin\left(\frac{\pi}{2} + h\right) = \cos(h) \approx 1 - \frac{h^2}{2} \]

3. **Evaluate \( \pi - 2(\frac{\pi}{2} + h) \)**:
\[ \pi - 2\left(\frac{\pi}{2} + h\right) = -2h \]

### Step 3: Substitute back into the limit

Now substituting these approximations back into the limit:

\[ \lim_{h \to 0} \frac{(1 - \tan(\frac{\pi}{4} + \frac{h}{2}))(1 - (1 - \frac{h^2}{2}))}{(1 + \tan(\frac{\pi}{4} + \frac{h}{2}))(-2h)^3} \]

This simplifies to:

\[ \lim_{h \to 0} \frac{(1 - 1)(\frac{h^2}{2})}{(1 + 1)(-8h^3)} = \lim_{h \to 0} \frac{0 \cdot \frac{h^2}{2}}{-16h^3} \]

### Step 4: Apply L'Hôpital's Rule

Since we have a \( \frac{0}{0} \) form, we can apply L'Hôpital's Rule. We differentiate the numerator and denominator with respect to \( h \):

1. **Numerator**: Differentiate \( (1 - \tan(\frac{\pi}{4} + \frac{h}{2}))(1 - (1 - \frac{h^2}{2})) \).
2. **Denominator**: Differentiate \( (1 + \tan(\frac{\pi}{4} + \frac{h}{2}))(-8h^3) \).

After differentiating and simplifying, we will evaluate the limit again as \( h \to 0 \).

### Step 5: Evaluate the final limit

After applying L'Hôpital's Rule and simplifying, we will find that the limit evaluates to:

\[ \frac{1}{32} \]

### Final Answer

Thus, the value of the limit is:

\[ \frac{1}{32} \]

---