The value of `lim _( x to (pi)/(4))(int_(2 ) ^(cosec ^(2)x)tg (t )dt)/(x ^(2)-(pi^(2))/(16))` is:

Options

A. `(2)/(pi) g (2)`
B. `-(4)/(pi) g (2)`
C. `-(16)/(pi) g (2)`
D. `-(16)/(pi) g (2)`

Answer

To solve the limit

\[ \lim_{x \to \frac{\pi}{4}} \frac{\int_{2}^{\csc^2 x} t g(t) dt}{x^2 - \frac{\pi^2}{16}}, \]

we start by evaluating the limit as \( x \) approaches \( \frac{\pi}{4} \).

### Step 1: Evaluate the denominator
First, we calculate the denominator:

\[ x^2 - \frac{\pi^2}{16}. \]

Substituting \( x = \frac{\pi}{4} \):

\[ \left( \frac{\pi}{4} \right)^2 - \frac{\pi^2}{16} = \frac{\pi^2}{16} - \frac{\pi^2}{16} = 0. \]

### Step 2: Evaluate the numerator
Next, we evaluate the numerator:

\[ \int_{2}^{\csc^2\left(\frac{\pi}{4}\right)} t g(t) dt. \]

Since \( \csc\left(\frac{\pi}{4}\right) = \sqrt{2} \), we have:

\[ \int_{2}^{\csc^2\left(\frac{\pi}{4}\right)} t g(t) dt = \int_{2}^{2} t g(t) dt = 0. \]

### Step 3: Determine the indeterminate form
Now, we have both the numerator and denominator approaching 0, giving us the indeterminate form \( \frac{0}{0} \). To resolve this, we can apply L'Hôpital's Rule.

### Step 4: Apply L'Hôpital's Rule
According to L'Hôpital's Rule, we differentiate the numerator and denominator separately.

#### Differentiate the numerator
Using Leibniz's rule for differentiation under the integral sign:

\[ \frac{d}{dx} \int_{2}^{\csc^2 x} t g(t) dt = g(\csc^2 x) \cdot \frac{d}{dx}(\csc^2 x). \]

The derivative of \( \csc^2 x \) is:

\[ \frac{d}{dx}(\csc^2 x) = -2 \csc^2 x \cot x. \]

Thus, the derivative of the numerator becomes:

\[ g(\csc^2 x) \cdot (-2 \csc^2 x \cot x). \]

#### Differentiate the denominator
The derivative of the denominator \( x^2 - \frac{\pi^2}{16} \) is:

\[ \frac{d}{dx}(x^2 - \frac{\pi^2}{16}) = 2x. \]

### Step 5: Rewrite the limit
Now we can rewrite the limit using the derivatives:

\[ \lim_{x \to \frac{\pi}{4}} \frac{-2 g(\csc^2 x) \csc^2 x \cot x}{2x}. \]

### Step 6: Simplify the limit
We can cancel the 2's:

\[ \lim_{x \to \frac{\pi}{4}} \frac{-g(\csc^2 x) \csc^2 x \cot x}{x}. \]

### Step 7: Substitute \( x = \frac{\pi}{4} \)
Now substituting \( x = \frac{\pi}{4} \):

\[ \csc^2\left(\frac{\pi}{4}\right) = 2, \quad \cot\left(\frac{\pi}{4}\right) = 1. \]

Thus, we have:

\[ \lim_{x \to \frac{\pi}{4}} \frac{-g(2) \cdot 2 \cdot 1}{\frac{\pi}{4}} = \frac{-8 g(2)}{\pi}. \]

### Final Result
Therefore, the value of the limit is:

\[ -\frac{8 g(2)}{\pi}. \]