`lim_(xrarr1)(1-x)tan((pix)/2)` is equal to

Options

A. `(pi)/2`
B. `2/(pi)`
C. `-(pi)/2`
D. `-2/(pi)`

Explanation

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

### Step 1: Substitute \( t = 1 - x \)
Let \( t = 1 - x \). As \( x \to 1 \), \( t \to 0 \). Therefore, we can rewrite the limit in terms of \( t \):
\[ x = 1 - t \implies \lim_{x \to 1} (1 - x) \tan\left(\frac{\pi x}{2}\right) = \lim_{t \to 0} t \tan\left(\frac{\pi (1 - t)}{2}\right) \]

### Step 2: Simplify the argument of the tangent function
Now, simplify the argument of the tangent:
\[ \frac{\pi (1 - t)}{2} = \frac{\pi}{2} - \frac{\pi t}{2} \]
Thus, we can rewrite the limit as:
\[ \lim_{t \to 0} t \tan\left(\frac{\pi}{2} - \frac{\pi t}{2}\right) \]

### Step 3: Use the identity for tangent
Using the identity \( \tan\left(\frac{\pi}{2} - x\right) = \cot(x) \), we can rewrite the limit:
\[ \lim_{t \to 0} t \cot\left(\frac{\pi t}{2}\right) \]

### Step 4: Rewrite cotangent in terms of sine and cosine
Recall that \( \cot(x) = \frac{\cos(x)}{\sin(x)} \). Therefore, we have:
\[ \lim_{t \to 0} t \cdot \frac{\cos\left(\frac{\pi t}{2}\right)}{\sin\left(\frac{\pi t}{2}\right)} \]

### Step 5: Analyze the limit
As \( t \to 0 \), \( \cos\left(\frac{\pi t}{2}\right) \to 1 \) and \( \sin\left(\frac{\pi t}{2}\right) \) can be approximated by its Taylor series expansion:
\[ \sin\left(\frac{\pi t}{2}\right) \approx \frac{\pi t}{2} \]
Thus, the limit becomes:
\[ \lim_{t \to 0} t \cdot \frac{1}{\frac{\pi t}{2}} = \lim_{t \to 0} \frac{2t}{\pi t} = \lim_{t \to 0} \frac{2}{\pi} = \frac{2}{\pi} \]

### Final Answer
Therefore, the limit is:
\[ \lim_{x \to 1} (1 - x) \tan\left(\frac{\pi x}{2}\right) = \frac{2}{\pi} \]