The limit expression given in the image is:

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

Let's work through it step by step.

Step 1: Simplify the expression

Notice that the numerator contains the term $2 \sin x \cos x$, which can be rewritten using a well-known trigonometric identity: $2 \sin x \cos x = \sin(2x)$ So the numerator becomes: $1 - 2 \sin x \cos x = 1 - \sin(2x)$

Thus, the limit now looks like: $\lim_{{x \to \frac{\pi}{4}}} \frac{1 - \sin(2x)}{\cos x - \sin x}$

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

We will now evaluate the individual trigonometric functions at $x = \frac{\pi}{4}$:

• $\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$
• $\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$
• $\sin(2x) = \sin\left(\frac{\pi}{2}\right) = 1$

Substituting these values into the expression, the numerator becomes: $1 - \sin\left(\frac{\pi}{2}\right) = 1 - 1 = 0$ And the denominator becomes: $\cos\left(\frac{\pi}{4}\right) - \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2} = 0$

Thus, we get the indeterminate form $\frac{0}{0}$, which suggests we should apply L'Hôpital's Rule.

Step 3: Apply L'Hôpital's Rule

L'Hôpital's Rule states that for limits of the indeterminate form $\frac{0}{0}$, we can differentiate the numerator and the denominator separately and then take the limit.

Derivative of the numerator:

$\frac{d}{dx} \left( 1 - \sin(2x) \right) = -2 \cos(2x)$

Derivative of the denominator:

$\frac{d}{dx} \left( \cos x - \sin x \right) = -\sin x - \cos x$

Thus, the limit becomes: $\lim_{{x \to \frac{\pi}{4}}} \frac{-2 \cos(2x)}{-\sin x - \cos x}$

Step 4: Evaluate the new limit

At $x = \frac{\pi}{4}$:

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

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

Thus, the value of the limit is: $\boxed{0}$

Would you like more details on any of the steps? Here are 5 related questions to deepen your understanding:

1. How does L'Hôpital's Rule help resolve indeterminate forms like $\frac{0}{0}$?
2. What are other common trigonometric identities that simplify expressions?
3. How can we determine if a limit needs L'Hôpital's Rule without evaluating it?
4. What are alternative methods to solve limits involving trigonometric functions?
5. Can you use approximation techniques to estimate limits near indeterminate points?

Tip: Always check for indeterminate forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$ before applying L'Hôpital's Rule, as it’s only valid in those cases.