We are tasked with evaluating the limit:

$\lim_{x \to \frac{\pi}{4}} \frac{1 - \tan(x)}{\sin(x) - \cos(x)}$

Step 1: Substitute $x = \frac{\pi}{4}$

First, let’s substitute $x = \frac{\pi}{4}$ directly into the expression:

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

So, at $x = \frac{\pi}{4}$, the expression becomes:

$\frac{1 - 1}{\frac{\sqrt{2}}{2} - \frac{\sqrt{2}}{2}} = \frac{0}{0}$

This results in an indeterminate form $\frac{0}{0}$, so we need to apply L'Hopital's Rule, which allows us to differentiate the numerator and denominator separately and then evaluate the limit.

Step 2: Apply L'Hopital's Rule

We need to differentiate the numerator and the denominator:

• The numerator is $1 - \tan(x)$. The derivative of $1$ is 0, and the derivative of $-\tan(x)$ is $-\sec^2(x)$. So, the derivative of the numerator is:

  $-\sec^2(x)$

• The denominator is $\sin(x) - \cos(x)$. The derivative of $\sin(x)$ is $\cos(x)$, and the derivative of $-\cos(x)$ is $\sin(x)$. So, the derivative of the denominator is:

  $\cos(x) + \sin(x)$

Step 3: Evaluate the new limit

Now we evaluate the new limit:

$\lim_{x \to \frac{\pi}{4}} \frac{-\sec^2(x)}{\cos(x) + \sin(x)}$

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

So, the limit becomes:

$\frac{-2}{\sqrt{2}} = -\sqrt{2}$

Final Answer:

$\lim_{x \to \frac{\pi}{4}} \frac{1 - \tan(x)}{\sin(x) - \cos(x)} = -\sqrt{2}$

Would you like more details on any step? Here are some related questions you might find useful:

1. How do you apply L'Hopital's Rule in more complicated cases?
2. What are the conditions under which L'Hopital’s Rule can be used?
3. Can limits be solved using series expansions instead of L'Hopital’s Rule?
4. How do you calculate derivatives of trigonometric functions like $\sec^2(x)$?
5. What is the geometric interpretation of the limit problem?

Tip: Always check the form of your limit before applying L'Hopital’s Rule to ensure it’s indeterminate, such as $\frac{0}{0}$ or $\frac{\infty}{\infty}$.