To evaluate the limit, we are given the function:

$g(x) = \frac{\cos x - \sin x}{1 - 2 \sin^2 x}$

and need to find:

$\lim_{x \to \frac{\pi}{4}} g(x)$

Step 1: Simplify the Expression

We start by substituting $x = \frac{\pi}{4}$ directly to check for any indeterminate form:

1. For the numerator: $\cos \frac{\pi}{4} - \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = 0$.
2. For the denominator: $1 - 2 \sin^2 \frac{\pi}{4} = 1 - 2 \cdot \left( \frac{\sqrt{2}}{2} \right)^2 = 1 - 2 \cdot \frac{1}{2} = 1 - 1 = 0$.

This yields a $\frac{0}{0}$ indeterminate form, so we can apply L'Hôpital's Rule.

Step 2: Apply L'Hôpital's Rule

To apply L'Hôpital's Rule, we differentiate the numerator and the denominator with respect to $x$.

1. Differentiate the numerator: $\frac{d}{dx}(\cos x - \sin x) = -\sin x - \cos x$

2. Differentiate the denominator: $\frac{d}{dx}(1 - 2 \sin^2 x) = -4 \sin x \cos x = -2 \sin(2x)$

After applying L'Hôpital's Rule, we have:

$\lim_{x \to \frac{\pi}{4}} g(x) = \lim_{x \to \frac{\pi}{4}} \frac{-\sin x - \cos x}{-2 \sin(2x)}$

Step 3: Substitute $x = \frac{\pi}{4}$ Again

Now, substitute $x = \frac{\pi}{4}$:

1. For the numerator: $-\sin \frac{\pi}{4} - \cos \frac{\pi}{4} = -\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = -\sqrt{2}$.
2. For the denominator: $-2 \sin\left(2 \cdot \frac{\pi}{4}\right) = -2 \sin\left(\frac{\pi}{2}\right) = -2 \cdot 1 = -2$.

Thus,

$\lim_{x \to \frac{\pi}{4}} g(x) = \frac{-\sqrt{2}}{-2} = \frac{\sqrt{2}}{2}$

Final Answer

The correct answer is:

$\boxed{\frac{1}{\sqrt{2}}}$

or, in decimal form, this is equivalent to answer choice B.

Would you like more details or have any questions?

Related Questions:

1. How would we approach the limit if L'Hôpital's Rule didn't apply?
2. What are common indeterminate forms that lead to using L'Hôpital's Rule?
3. How do trigonometric identities help in simplifying limits?
4. How can we use Taylor series expansions for similar trigonometric limits?
5. What if we needed to evaluate $\lim_{x \to \frac{\pi}{2}} g(x)$?

Tip:

When dealing with trigonometric limits, checking for direct substitution and possible simplifications using trigonometric identities can often save time before jumping to L'Hôpital's Rule.