The value of `lim _(x to pi/4) (pi/4 int_2^(sec^2x)f(x)dx)/(x^2-pi^2/16)` is

Options

A. `2f(2)`
B. `sqrt2 f(2)`
C. `4f(2)`
D. `f(sqrt2)`

Answer

To solve the limit

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

we first need to analyze the expression as \( x \) approaches \( \frac{\pi}{4} \).

### Step 1: Evaluate the limit's form

As \( x \to \frac{\pi}{4} \), we have:

- \( \sec^2\left(\frac{\pi}{4}\right) = 2 \)
- Therefore, \( \int_{2}^{\sec^2 x} f(t) \, dt \) approaches \( \int_{2}^{2} f(t) \, dt = 0 \).
- The denominator \( x^2 - \frac{\pi^2}{16} \) also approaches \( \left(\frac{\pi}{4}\right)^2 - \frac{\pi^2}{16} = 0 \).

Thus, we have a \( \frac{0}{0} \) indeterminate form, which allows us to apply L'Hôpital's Rule.

### Step 2: Apply L'Hôpital's Rule

According to L'Hôpital's Rule, we differentiate the numerator and the denominator:

1. **Differentiate the numerator:**

The numerator is \( \frac{\pi}{4} \int_{2}^{\sec^2 x} f(t) \, dt \). We apply the Fundamental Theorem of Calculus and the chain rule:

\[ \frac{d}{dx}\left(\frac{\pi}{4} \int_{2}^{\sec^2 x} f(t) \, dt\right) = \frac{\pi}{4} f(\sec^2 x) \cdot \frac{d}{dx}(\sec^2 x). \]

The derivative of \( \sec^2 x \) is \( 2 \sec^2 x \tan x \). Thus, we have:

\[ \frac{d}{dx}\left(\frac{\pi}{4} \int_{2}^{\sec^2 x} f(t) \, dt\right) = \frac{\pi}{4} f(\sec^2 x) \cdot (2 \sec^2 x \tan x). \]

2. **Differentiate the denominator:**

The denominator \( x^2 - \frac{\pi^2}{16} \) differentiates to \( 2x \).

### Step 3: Rewrite the limit

Now we rewrite the limit using the derivatives:

\[ \lim_{x \to \frac{\pi}{4}} \frac{\frac{\pi}{4} f(\sec^2 x) \cdot (2 \sec^2 x \tan x)}{2x}. \]

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

Now we substitute \( x = \frac{\pi}{4} \):

- \( \sec^2\left(\frac{\pi}{4}\right) = 2 \)
- \( \tan\left(\frac{\pi}{4}\right) = 1 \)
- \( 2x = 2 \cdot \frac{\pi}{4} = \frac{\pi}{2} \)

Thus, we have:

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

### Step 5: Simplify the expression

This simplifies to:

\[ \frac{\pi f(2)}{2} \cdot \frac{2}{\pi} = 2 f(2). \]

### Final Answer

Thus, the value of the limit is:

\[ \boxed{2 f(2)}. \]