The value of `lim_(xto1) (2-x)^(tan((pix)/(2)))` is

Options

A. `a=1//4`
B. `b=3//4`
C. `L=-1//32`
D. `L=1//32`

Explanation

To find the limit \( \lim_{x \to 1} (2 - x)^{\tan\left(\frac{\pi x}{2}\right)} \), we can follow these steps:

### Step 1: Identify the form of the limit
First, we substitute \( x = 1 \) into the expression:
\[ 2 - x \to 2 - 1 = 1 \]
\[ \tan\left(\frac{\pi x}{2}\right) \to \tan\left(\frac{\pi \cdot 1}{2}\right) = \tan\left(\frac{\pi}{2}\right) \to \infty \]
Thus, we have the form \( 1^{\infty} \), which is an indeterminate form.

**Hint**: Recognize that \( 1^{\infty} \) is an indeterminate form, and we need to manipulate the expression to resolve it.

### Step 2: Rewrite the limit using exponential form
We can rewrite the limit in exponential form:
\[ (2 - x)^{\tan\left(\frac{\pi x}{2}\right)} = e^{\tan\left(\frac{\pi x}{2}\right) \ln(2 - x)} \]
So we need to evaluate:
\[ \lim_{x \to 1} \tan\left(\frac{\pi x}{2}\right) \ln(2 - x) \]

**Hint**: Use the property of logarithms to express the limit in a more manageable form.

### Step 3: Evaluate the limit of the exponent
Now we need to find:
\[ \lim_{x \to 1} \tan\left(\frac{\pi x}{2}\right) \ln(2 - x) \]
As \( x \to 1 \):
- \( \ln(2 - x) \to \ln(1) = 0 \)
- \( \tan\left(\frac{\pi x}{2}\right) \to \infty \)

This gives us the form \( \infty \cdot 0 \), which is also indeterminate. We can rewrite it as:
\[ \lim_{x \to 1} \frac{\ln(2 - x)}{\cot\left(\frac{\pi x}{2}\right)} \]

**Hint**: Convert the product into a quotient to apply L'Hôpital's rule.

### Step 4: Apply L'Hôpital's Rule
Now we apply L'Hôpital's Rule, since both the numerator and denominator approach 0 as \( x \to 1 \):
\[ \text{Differentiate the numerator: } \frac{d}{dx}[\ln(2 - x)] = -\frac{1}{2 - x} \]
\[ \text{Differentiate the denominator: } \frac{d}{dx}[\cot\left(\frac{\pi x}{2}\right)] = -\frac{\pi}{2} \csc^2\left(\frac{\pi x}{2}\right) \]
Thus, we have:
\[ \lim_{x \to 1} \frac{-\frac{1}{2 - x}}{-\frac{\pi}{2} \csc^2\left(\frac{\pi x}{2}\right)} = \lim_{x \to 1} \frac{2}{\pi (2 - x) \csc^2\left(\frac{\pi x}{2}\right)} \]

**Hint**: Remember to evaluate the limit as \( x \to 1 \) carefully.

### Step 5: Evaluate the limit
As \( x \to 1 \):
- \( 2 - x \to 1 \)
- \( \csc^2\left(\frac{\pi x}{2}\right) \to \csc^2\left(\frac{\pi}{2}\right) \to 1 \)

Thus, we get:
\[ \lim_{x \to 1} \frac{2}{\pi (1)(1)} = \frac{2}{\pi} \]

### Step 6: Final result
Now substituting back into the exponential:
\[ \lim_{x \to 1} (2 - x)^{\tan\left(\frac{\pi x}{2}\right)} = e^{\frac{2}{\pi}} \]

**Final Answer**:
\[ \lim_{x \to 1} (2 - x)^{\tan\left(\frac{\pi x}{2}\right)} = e^{\frac{2}{\pi}} \]