Evaluate`lim_(xto pi//4) (2-tanx)^(1//1n(tanx))`

Options

A. 16
B. 8
C. 4
D. 2

Explanation

To evaluate the limit

\[ \lim_{x \to \frac{\pi}{4}} (2 - \tan x)^{\frac{1}{\ln(\tan x)}} \]

we will follow these steps:

### Step 1: Substitute \( x = \frac{\pi}{4} \)

First, we substitute \( x = \frac{\pi}{4} \) into the expression:

\[ \tan\left(\frac{\pi}{4}\right) = 1 \]

Thus, we have:

\[ 2 - \tan\left(\frac{\pi}{4}\right) = 2 - 1 = 1 \]

And since \( \ln(\tan\left(\frac{\pi}{4}\right)) = \ln(1) = 0 \), we find that the expression becomes:

\[ 1^{\frac{1}{0}} \quad \text{(which is an indeterminate form of type } 1^{\infty}\text{)} \]

### Step 2: Rewrite the expression

Since we have an indeterminate form \( 1^{\infty} \), we can use the property that if \( f(x) \to 1 \) and \( g(x) \to \infty \), then:

\[ \lim_{x \to a} f(x)^{g(x)} = e^{\lim_{x \to a} g(x)(f(x) - 1)} \]

Here, we can let:

- \( f(x) = 2 - \tan x \)
- \( g(x) = \frac{1}{\ln(\tan x)} \)

Thus, we need to compute:

\[ \lim_{x \to \frac{\pi}{4}} \frac{1}{\ln(\tan x)} \cdot (2 - \tan x - 1) = \lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{\ln(\tan x)} \]

### Step 3: Simplify the limit

Now we rewrite the limit:

\[ \lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{\ln(\tan x)} \]

As \( x \to \frac{\pi}{4} \), both the numerator \( 1 - \tan x \) and the denominator \( \ln(\tan x) \) approach 0, giving us another indeterminate form \( \frac{0}{0} \). We can apply L'Hôpital's Rule:

### Step 4: Apply L'Hôpital's Rule

Differentiating the numerator and denominator:

- The derivative of \( 1 - \tan x \) is \( -\sec^2 x \)
- The derivative of \( \ln(\tan x) \) is \( \frac{\sec^2 x}{\tan x} \)

Thus, we have:

\[ \lim_{x \to \frac{\pi}{4}} \frac{-\sec^2 x}{\frac{\sec^2 x}{\tan x}} = \lim_{x \to \frac{\pi}{4}} -\tan x \]

### Step 5: Evaluate the limit

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

\[ -\tan\left(\frac{\pi}{4}\right) = -1 \]

### Step 6: Final expression

Now we can substitute back into our earlier expression:

\[ \lim_{x \to \frac{\pi}{4}} (2 - \tan x)^{\frac{1}{\ln(\tan x)}} = e^{-1} \]

Thus, the final answer is:

\[ \boxed{\frac{1}{e}} \]