Evaluate the following limits :
`Lim_(x to pi/4) (1- tan x)/(x - pi/4 )`

Answer

To evaluate the limit \( \lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{x - \frac{\pi}{4}} \), we will follow these steps:

### Step 1: Change of Variable
Let \( \theta = x - \frac{\pi}{4} \). Then as \( x \to \frac{\pi}{4} \), \( \theta \to 0 \). We can rewrite \( x \) as \( \theta + \frac{\pi}{4} \).

### Step 2: Rewrite the Limit
Now, substitute \( x \) in the limit:
\[ \lim_{\theta \to 0} \frac{1 - \tan(\theta + \frac{\pi}{4})}{\theta} \]

### Step 3: Use the Tan Addition Formula
Using the tangent addition formula:
\[ \tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]
where \( A = \theta \) and \( B = \frac{\pi}{4} \), we have:
\[ \tan(\theta + \frac{\pi}{4}) = \frac{\tan \theta + 1}{1 - \tan \theta} \]
Substituting this back into the limit gives:
\[ \lim_{\theta \to 0} \frac{1 - \frac{\tan \theta + 1}{1 - \tan \theta}}{\theta} \]

### Step 4: Simplify the Expression
Now simplify the expression:
\[ 1 - \frac{\tan \theta + 1}{1 - \tan \theta} = \frac{(1 - \tan \theta) - (\tan \theta + 1)}{1 - \tan \theta} = \frac{1 - \tan \theta - \tan \theta - 1}{1 - \tan \theta} = \frac{-2 \tan \theta}{1 - \tan \theta} \]
Thus, the limit becomes:
\[ \lim_{\theta \to 0} \frac{-2 \tan \theta}{\theta (1 - \tan \theta)} \]

### Step 5: Use the Limit Identity
We know that \( \lim_{\theta \to 0} \frac{\tan \theta}{\theta} = 1 \). Therefore:
\[ \lim_{\theta \to 0} \frac{-2 \tan \theta}{\theta} = -2 \]
And \( \lim_{\theta \to 0} (1 - \tan \theta) = 1 - 0 = 1 \).

### Step 6: Final Limit Calculation
Putting it all together:
\[ \lim_{\theta \to 0} \frac{-2 \tan \theta}{\theta (1 - \tan \theta)} = \frac{-2}{1} = -2 \]

### Final Answer
Thus, the limit is:
\[ \boxed{-2} \]