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To solve this problem, we need to analyze the function given and evaluate the limit:

### Problem Analysis
We have the function:
\[
g(x) = \frac{2\cos^2 x - 1}{\cos x - \sin x}
\]
and we are asked to find \(\lim_{x \to \frac{\pi}{4}} g(x)\).

### Step 1: Simplify the Expression
We know that:
\[
2\cos^2 x - 1 = \cos(2x)
\]
Thus, we can rewrite \(g(x)\) as:
\[
g(x) = \frac{\cos(2x)}{\cos x - \sin x}
\]

### Step 2: Substitute \(x = \frac{\pi}{4}\)
When \(x = \frac{\pi}{4}\):
- \(\cos\left(\frac{\pi}{4}\right) = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}\).
- So \(\cos x - \sin x = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = 0\), making the denominator zero.
- Similarly, \(2x = \frac{\pi}{2}\), so \(\cos(2x) = \cos\left(\frac{\pi}{2}\right) = 0\), making the numerator zero as well.

Since we have a \(\frac{0}{0}\) indeterminate form, we can apply **L'Hôpital's Rule** to evaluate the limit.

### Step 3: Apply L'Hôpital's Rule
Differentiating the numerator and denominator with respect to \(x\):
1. **Numerator:** The derivative of \(\cos(2x)\) is \(-2\sin(2x)\).
2. **Denominator:** The derivative of \(\cos x - \sin x\) is \(-\sin x - \cos x\).

So we get:
\[
\lim_{x \to \frac{\pi}{4}} g(x) = \lim_{x \to \frac{\pi}{4}} \frac{-2\sin(2x)}{-\sin x - \cos x} = \lim_{x \to \frac{\pi}{4}} \frac{2\sin(2x)}{\sin x + \cos x}
\]

### Step 4: Substitute \(x = \frac{\pi}{4}\) Again
When \(x = \frac{\pi}{4}\):
- \(\sin(2x) = \sin\left(\frac{\pi}{2}\right) = 1\),
- \(\sin x + \cos x = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2}\).

Thus:
\[
\lim_{x \to \frac{\pi}{4}} g(x) = \frac{2 \cdot 1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}
\]

### Conclusion
The correct answer is:
\[
\boxed{\sqrt{2}}
\]

Looking at the answer choices, this matches **Option B**:

\[
\lim_{x \to \frac{\pi}{4}} (\cos x + \sin x)
\]

Let me know if you want details or have any questions.

---

Here are five related questions to expand your understanding:

1. How does L'Hôpital's Rule help in evaluating limits involving indeterminate forms?
2. What other trigonometric identities might be useful in simplifying expressions like \(2\cos^2 x - 1\)?
3. How do we know when to apply L'Hôpital's Rule versus algebraic simplification?
4. What is the geometric interpretation of \(\lim_{x \to \frac{\pi}{4}} (\cos x + \sin x)\)?
5. How would the result change if the function had a different trigonometric expression in the denominator?

**Tip:** Remember to check for indeterminate forms before applying L'Hôpital's Rule.

If g is the function defined by g(x) = (2cos^2(x) - 1) / (cos(x) - sin(x)), then lim(x → π/4) g(x) is equivalent to which of the following?

Learn how to evaluate the limit of the function g(x) = (2cos^2(x) - 1) / (cos(x) - sin(x)) as x approaches π/4. This problem involves the use of trigonometric identities and L'Hôpital's Rule to simplify an indeterminate form.

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