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: Evaluate the limit directly
First, we substitute \( x = 2\pi \) into the expression:

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

Thus, we have:

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

And,

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

This gives us the form \( \frac{0}{0} \), which is indeterminate.

**Hint for Step 1:** Check the values of the functions at the limit point to see if you get an indeterminate form.

### Step 2: Apply L'Hôpital's Rule
Since we have the 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.

### Step 3: Differentiate the numerator
Let \( y = (\sec x)^{\sec x} \). Taking the natural logarithm on both sides:

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

Differentiating both sides with respect to \( x \):

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

Thus,

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

Substituting back for \( y \):

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

### Step 4: Differentiate the denominator
The derivative of \( \ln(\sec x) \) is:

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

### Step 5: Apply L'Hôpital's Rule
Now we can rewrite our limit using L'Hôpital's Rule:

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

### Step 6: Simplify the expression
Cancelling \( \sec x \tan x \) from the numerator and denominator gives:

\[ \lim_{x \to 2\pi} \left( (\sec x)^{\sec x} \left( 1 + \ln(\sec x) \right) \right) \]

### Step 7: Substitute \( x = 2\pi \)
Now substituting \( x = 2\pi \):

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

And since \( \ln(\sec(2\pi)) = 0 \):

\[ 1 \cdot (1 + 0) = 1 \]

### Step 8: Final limit evaluation
Thus, the limit evaluates to:

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

### Conclusion
The value of the limit is:

\[ \boxed{-1} \]