The given expression is:

$\lim_{x \to \frac{\pi}{4}} \frac{1 - 2\sin x \cos x}{\sin x - \cos x}$

Step-by-Step Solution:

1. Simplify the numerator: Using the double-angle identity for sine, $2 \sin x \cos x = \sin(2x)$, we can rewrite the numerator: $1 - 2 \sin x \cos x = 1 - \sin(2x)$ So, the expression becomes: $\lim_{x \to \frac{\pi}{4}} \frac{1 - \sin(2x)}{\sin x - \cos x}$

2. Evaluate at $x = \frac{\pi}{4}$:

   • $\sin \left( \frac{\pi}{4} \right) = \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2}$
   • Substituting into the denominator: $\sin\left(\frac{\pi}{4}\right) - \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = 0$
   • Substituting into the numerator: $1 - \sin\left(2 \cdot \frac{\pi}{4}\right) = 1 - \sin\left(\frac{\pi}{2}\right) = 1 - 1 = 0$ So, both the numerator and denominator evaluate to 0, giving us an indeterminate form $\frac{0}{0}$. This suggests we should apply L'Hopital's Rule.

3. Apply L'Hopital's Rule: To resolve the indeterminate form, we differentiate the numerator and the denominator with respect to $x$:

   • The derivative of the numerator $1 - \sin(2x)$ is: $-2 \cos(2x)$
   • The derivative of the denominator $\sin x - \cos x$ is: $\cos x + \sin x$

   So, applying L'Hopital's Rule, we now compute the limit of: $\lim_{x \to \frac{\pi}{4}} \frac{-2 \cos(2x)}{\cos x + \sin x}$

4. Evaluate the new expression at $x = \frac{\pi}{4}$:

   • $\cos\left(2 \cdot \frac{\pi}{4}\right) = \cos\left(\frac{\pi}{2}\right) = 0$
   • $\cos\left(\frac{\pi}{4}\right) + \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = \sqrt{2}$

   Therefore, the expression becomes: $\frac{-2 \cdot 0}{\sqrt{2}} = 0$

Final Answer:

$\lim_{x \to \frac{\pi}{4}} \frac{1 - 2 \sin x \cos x}{\sin x - \cos x} = 0$

Would you like more details on any step or further clarification?

Here are 5 related questions you might be interested in:

1. What is L'Hopital's Rule, and when is it used?
2. How can trigonometric identities simplify limits involving sine and cosine?
3. What is the double-angle identity, and how does it help in solving limits?
4. How would this problem change if we were approaching a different value for $x$?
5. What are some common forms of indeterminate limits?

Tip: Always check for indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ before applying L'Hopital's Rule!