Evaluate the following limits :
`Lim_(xto 1) (1-x) (tan. (pix)/2)`

Answer

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

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

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

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

### Step 4: Rewrite cotangent in terms of tangent
Recall that \( \cot(\theta) = \frac{1}{\tan(\theta)} \):
\[ \lim_{y \to 0} y \cot\left(\frac{\pi y}{2}\right) = \lim_{y \to 0} \frac{y}{\tan\left(\frac{\pi y}{2}\right)} \]

### Step 5: Apply the limit identity
We know from the limit identity that \( \lim_{\theta \to 0} \frac{\tan(\theta)}{\theta} = 1 \). Thus:
\[ \lim_{y \to 0} \frac{y}{\tan\left(\frac{\pi y}{2}\right)} = \lim_{y \to 0} \frac{y}{\frac{\pi y}{2}} = \lim_{y \to 0} \frac{2}{\pi} = \frac{2}{\pi} \]

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