Evaluate `lim_(x to (pi)/(4)) ("sin" x - cos x)/(x - (pi)/(4))`

Answer

To evaluate the limit

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

we can follow these steps:

### Step 1: Substitute \( x = h + \frac{\pi}{4} \)

Let \( h = x - \frac{\pi}{4} \). As \( x \to \frac{\pi}{4} \), \( h \to 0 \). Thus, we rewrite the limit:

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

### Step 2: Use the angle addition formulas

Using the angle addition formulas for sine and cosine:

\[ \sin(h + \frac{\pi}{4}) = \sin h \cos \frac{\pi}{4} + \cos h \sin \frac{\pi}{4}, \]
\[ \cos(h + \frac{\pi}{4}) = \cos h \cos \frac{\pi}{4} - \sin h \sin \frac{\pi}{4}. \]

Substituting these into the limit gives us:

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

### Step 3: Simplify the expression

Simplifying the numerator:

\[ = \lim_{h \to 0} \frac{\frac{1}{\sqrt{2}}(\sin h + \sin h) - \frac{1}{\sqrt{2}}(\cos h - \cos h)}{h} \]

\[ = \lim_{h \to 0} \frac{\frac{2}{\sqrt{2}} \sin h}{h}. \]

### Step 4: Factor out constants

We can factor out the constant \( \frac{2}{\sqrt{2}} \):

\[ = \frac{2}{\sqrt{2}} \lim_{h \to 0} \frac{\sin h}{h}. \]

### Step 5: Apply the limit property

Using the known limit \( \lim_{h \to 0} \frac{\sin h}{h} = 1 \):

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

### Final Answer

Thus, the limit evaluates to:

\[ \lim_{x \to \frac{\pi}{4}} \frac{\sin x - \cos x}{x - \frac{\pi}{4}} = \sqrt{2}. \]

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