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.