https://r2.erweima.ai/mathbot/image/1727217156702ux0dqu3f.jpg

The image presents a limit problem involving trigonometric functions, specifically:

\[
\lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{\sin x - \cos x}
\]

The problem then discusses simplifying this expression through operations and obtaining a specific result.

### Problem Breakdown

To solve the limit:

\[
\lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{\sin x - \cos x}
\]

we need to evaluate each trigonometric function at \(x = \frac{\pi}{4}\):

- \(\sin\left(\frac{\pi}{4}\right) = \cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\)
- \(\tan\left(\frac{\pi}{4}\right) = 1\)

Substituting these values into the expression:

\[
\frac{1 - \tan\left(\frac{\pi}{4}\right)}{\sin\left(\frac{\pi}{4}\right) - \cos\left(\frac{\pi}{4}\right)} = \frac{1 - 1}{\frac{1}{\sqrt{2}} - \frac{1}{\sqrt{2}}}
\]

This results in:

\[
\frac{0}{0}
\]

This is an indeterminate form, so we can apply **L'Hopital's Rule**, which involves taking the derivative of the numerator and denominator.

#### Applying L'Hopital's Rule:

1. Derivative of the numerator \(1 - \tan x\):
   \[
   \frac{d}{dx}(1 - \tan x) = -\sec^2 x
   \]
2. Derivative of the denominator \(\sin x - \cos x\):
   \[
   \frac{d}{dx}(\sin x - \cos x) = \cos x + \sin x
   \]

So the limit becomes:

\[
\lim_{x \to \frac{\pi}{4}} \frac{-\sec^2 x}{\cos x + \sin x}
\]

Substituting \(x = \frac{\pi}{4}\):

- \(\sec^2\left(\frac{\pi}{4}\right) = 2\)
- \(\cos\left(\frac{\pi}{4}\right) + \sin\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \sqrt{2}\)

Thus, the limit becomes:

\[
\frac{-2}{\sqrt{2}} = \frac{-2}{\sqrt{2}} = -\sqrt{2}
\]

### Conclusion

The correct transformation is:

\[
\lim_{x \to \frac{\pi}{4}} \frac{-1}{\sin x} \quad \text{and the result is} \quad \frac{-2}{\sqrt{2}}
\]

This matches the last option in the image, which is highlighted.

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Do you want more details or have any questions about this?

Here are 5 related questions for further exploration:

1. How is L'Hopital's Rule applied to limits with indeterminate forms?
2. What are other ways to solve limits involving trigonometric functions without using L'Hopital's Rule?
3. Why does \( \frac{0}{0} \) indicate an indeterminate form?
4. Can the same process be applied if the angle was different from \( \frac{\pi}{4} \)?
5. What is the significance of trigonometric identities in simplifying limits?

**Tip**: Always check for indeterminate forms when solving limits; if you encounter \( \frac{0}{0} \), consider using L'Hopital's Rule.

En el proceso para hallar el resultado de lim x→π/4 (1−tan(x))/(sin(x)−cos(x)), se realizan operaciones hasta transformar la expresión y obtener la forma deseada.

Explore how to solve the limit lim x→π/4 (1−tan(x))/(sin(x)−cos(x)) by applying L'Hopital's Rule. The problem involves trigonometric functions, their derivatives, and simplifications, resulting in a final expression.

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