Evaluate `lim_(xto1) (1-x)"tan"(pix)/(2).`

Explanation

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

### Step 1: Rewrite the limit
We start with the limit expression:
\[ \lim_{x \to 1} (1 - x) \tan\left(\frac{\pi x}{2}\right) \]

### Step 2: Identify the form of the limit
As \( x \to 1 \), \( 1 - x \to 0 \) and \( \tan\left(\frac{\pi x}{2}\right) \to \tan\left(\frac{\pi}{2}\right) \), which is undefined. However, we can analyze the behavior of \( \tan\left(\frac{\pi x}{2}\right) \) as \( x \) approaches 1.

### Step 3: Use the identity for tangent
Recall that \( \tan\left(\frac{\pi x}{2}\right) \) approaches infinity as \( x \to 1 \). To handle this, we can rewrite the tangent function using the identity:
\[ \tan\left(\frac{\pi x}{2}\right) = \frac{\sin\left(\frac{\pi x}{2}\right)}{\cos\left(\frac{\pi x}{2}\right)} \]
As \( x \to 1 \), \( \cos\left(\frac{\pi x}{2}\right) \to 0 \) and \( \sin\left(\frac{\pi x}{2}\right) \to 1 \).

### Step 4: Rewrite the limit using sine and cosine
Thus, we can express the limit as:
\[ \lim_{x \to 1} (1 - x) \cdot \frac{\sin\left(\frac{\pi x}{2}\right)}{\cos\left(\frac{\pi x}{2}\right)} \]

### Step 5: Substitute \( y = 1 - x \)
Let \( y = 1 - x \), then as \( x \to 1 \), \( y \to 0 \). We rewrite \( x \) in terms of \( y \):
\[ x = 1 - y \]
Thus, the limit becomes:
\[ \lim_{y \to 0} y \cdot \tan\left(\frac{\pi (1 - y)}{2}\right) = \lim_{y \to 0} y \cdot \tan\left(\frac{\pi}{2} - \frac{\pi y}{2}\right) \]

### Step 6: Use the identity for tangent
Using the identity \( \tan\left(\frac{\pi}{2} - \theta\right) = \cot(\theta) \), we have:
\[ \tan\left(\frac{\pi}{2} - \frac{\pi y}{2}\right) = \cot\left(\frac{\pi y}{2}\right) \]

### Step 7: Rewrite the limit
Now the limit can be rewritten as:
\[ \lim_{y \to 0} y \cdot \cot\left(\frac{\pi y}{2}\right) \]

### Step 8: Use the cotangent identity
Recall that \( \cot(z) = \frac{\cos(z)}{\sin(z)} \). As \( y \to 0 \), we can use the approximation:
\[ \cot\left(\frac{\pi y}{2}\right) \approx \frac{2}{\pi y} \]
Thus, the limit becomes:
\[ \lim_{y \to 0} y \cdot \frac{2}{\pi y} = \lim_{y \to 0} \frac{2}{\pi} = \frac{2}{\pi} \]

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