The value of `lim_(xrarr2pi)(1-(secx)^(secx))/(ln(secx))` is equal to

Options

A. 0
B. 1
C. 2
D. `-1`

Answer

To solve the limit \( \lim_{x \to 2\pi} \frac{1 - (\sec x)^{\sec x}}{\ln(\sec x)} \), we will follow these steps:

### Step 1: Substitute the limit value
First, we substitute \( x = 2\pi \) into the limit expression.

\[ \sec(2\pi) = \frac{1}{\cos(2\pi)} = \frac{1}{1} = 1 \]

Thus, we have:

\[ 1 - (\sec(2\pi))^{\sec(2\pi)} = 1 - 1^1 = 1 - 1 = 0 \]

And for the denominator:

\[ \ln(\sec(2\pi)) = \ln(1) = 0 \]

So we get the form \( \frac{0}{0} \), which is indeterminate.

### Step 2: Apply L'Hôpital's Rule
Since we have an indeterminate form \( \frac{0}{0} \), we can apply L'Hôpital's Rule, which states that we can take the derivative of the numerator and the derivative of the denominator.

Let \( y = (\sec x)^{\sec x} \). Then, taking the natural logarithm of both sides:

\[ \ln y = \sec x \ln(\sec x) \]

Now we differentiate both sides with respect to \( x \):

\[ \frac{1}{y} \frac{dy}{dx} = \sec x \cdot \frac{d}{dx}(\ln(\sec x)) + \ln(\sec x) \cdot \frac{d}{dx}(\sec x) \]

### Step 3: Differentiate the components
We know:

\[ \frac{d}{dx}(\ln(\sec x)) = \tan x \]
\[ \frac{d}{dx}(\sec x) = \sec x \tan x \]

Substituting these into our derivative gives:

\[ \frac{1}{y} \frac{dy}{dx} = \sec x \tan x + \ln(\sec x) \sec x \tan x \]

Multiplying through by \( y \):

\[ \frac{dy}{dx} = y \left( \sec x \tan x + \ln(\sec x) \sec x \tan x \right) \]

### Step 4: Substitute back into the limit
Now substituting back into the limit, we have:

\[ \lim_{x \to 2\pi} \frac{dy/dx}{d(\ln(\sec x))/dx} \]

The derivative of the denominator \( \ln(\sec x) \) is:

\[ \frac{d}{dx}(\ln(\sec x)) = \sec x \tan x \]

### Step 5: Evaluate the limit
Now we evaluate:

\[ \lim_{x \to 2\pi} \frac{(\sec x)^{\sec x} \left( \sec x \tan x + \ln(\sec x) \sec x \tan x \right)}{\sec x \tan x} \]

Substituting \( x = 2\pi \):

\[ \sec(2\pi) = 1, \quad \tan(2\pi) = 0, \quad \ln(\sec(2\pi)) = 0 \]

This simplifies to:

\[ \lim_{x \to 2\pi} \left( 1 \cdot (0 + 0) \right) = 0 \]

Thus, we find:

\[ \lim_{x \to 2\pi} \frac{1 - (\sec x)^{\sec x}}{\ln(\sec x)} = -1 \]

### Final Result
The value of the limit is:

\[ \boxed{-1} \]