Evaluate the following limit :
`Lim_(theta to pi/4) (sin theta -cos theta)/(theta-1/4pi)`

Answer

To evaluate the limit

\[ \lim_{\theta \to \frac{\pi}{4}} \frac{\sin \theta - \cos \theta}{\theta - \frac{\pi}{4}}, \]

we can follow these steps:

### Step 1: Rewrite the Limit
We rewrite the limit in terms of a new variable \( x \):
Let \( x = \theta - \frac{\pi}{4} \). Then, as \( \theta \to \frac{\pi}{4} \), \( x \to 0 \). Thus, we can rewrite the limit as:

\[ \lim_{x \to 0} \frac{\sin\left(x + \frac{\pi}{4}\right) - \cos\left(x + \frac{\pi}{4}\right)}{x}. \]

### Step 2: Use Angle Addition Formulas
Using the angle addition formulas for sine and cosine, we have:

\[ \sin\left(x + \frac{\pi}{4}\right) = \sin x \cos\frac{\pi}{4} + \cos x \sin\frac{\pi}{4} \]
\[ \cos\left(x + \frac{\pi}{4}\right) = \cos x \cos\frac{\pi}{4} - \sin x \sin\frac{\pi}{4}. \]

Substituting these into the limit gives us:

\[ \lim_{x \to 0} \frac{\left(\sin x \cdot \frac{1}{\sqrt{2}} + \cos x \cdot \frac{1}{\sqrt{2}}\right) - \left(\cos x \cdot \frac{1}{\sqrt{2}} - \sin x \cdot \frac{1}{\sqrt{2}}\right)}{x}. \]

### Step 3: Simplify the Expression
Now simplifying the numerator:

\[ \sin x \cdot \frac{1}{\sqrt{2}} + \cos x \cdot \frac{1}{\sqrt{2}} - \cos x \cdot \frac{1}{\sqrt{2}} + \sin x \cdot \frac{1}{\sqrt{2}} = 2\sin x \cdot \frac{1}{\sqrt{2}}. \]

Thus, we can rewrite the limit as:

\[ \lim_{x \to 0} \frac{2 \sin x \cdot \frac{1}{\sqrt{2}}}{x}. \]

### Step 4: Apply the Limit
We know that

\[ \lim_{x \to 0} \frac{\sin x}{x} = 1. \]

So we can substitute this into our limit:

\[ = 2 \cdot \frac{1}{\sqrt{2}} \cdot 1 = \frac{2}{\sqrt{2}} = \sqrt{2}. \]

### Final Answer
Thus, the limit evaluates to:

\[ \sqrt{2}. \]