The problem involves calculating the limit of $\lim_{x \to \frac{\pi}{4}} \left[ (\tan x - 1) \sec(2x) \right]$. Let’s analyze and solve this step by step.

Step 1: Analyze the expression

We aim to compute: $\lim_{x \to \frac{\pi}{4}} \left[ (\tan x - 1) \sec(2x) \right]$

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

• $\tan\left(\frac{\pi}{4}\right) = 1$, so $(\tan x - 1) \to 0$,
• $\sec(2x) = \sec\left(\frac{\pi}{2}\right)$, which diverges ($\sec(\frac{\pi}{2}) = \infty$).

This creates an indeterminate form $0 \cdot \infty$. To resolve this, we rewrite the expression.

Step 2: Rewrite the expression

We rewrite $\sec(2x)$ as $\frac{1}{\cos(2x)}$, giving: $(\tan x - 1) \sec(2x) = \frac{\tan x - 1}{\cos(2x)}$

Now, analyze the new form:

• As $x \to \frac{\pi}{4}$, $\tan x - 1 \to 0$,
• $\cos(2x) \to \cos\left(\frac{\pi}{2}\right) = 0$.

This creates an indeterminate form $\frac{0}{0}$, which can be resolved using L'Hôpital's Rule.

Step 3: Apply L’Hôpital’s Rule

Using L'Hôpital’s Rule, differentiate the numerator and denominator:

• Numerator: $\frac{d}{dx}(\tan x - 1) = \sec^2 x$,
• Denominator: $\frac{d}{dx}(\cos(2x)) = -2\sin(2x)$.

Thus, the limit becomes: $\lim_{x \to \frac{\pi}{4}} \frac{\tan x - 1}{\cos(2x)} = \lim_{x \to \frac{\pi}{4}} \frac{\sec^2 x}{-2\sin(2x)}$

Step 4: Simplify and substitute

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

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

Substitute these values into the expression: $\lim_{x \to \frac{\pi}{4}} \frac{\sec^2 x}{-2\sin(2x)} = \frac{2}{-2 \cdot 1} = -1.$

Final Answer:

$\boxed{-1}$

Would you like further clarification on any step or additional examples? Here are 5 related questions you might explore:

1. How do you identify indeterminate forms like $0 \cdot \infty$ or $\frac{0}{0}$?
2. How does L'Hôpital's Rule work for different forms like $\frac{0}{0}$ or $\frac{\infty}{\infty}$?
3. Can limits involving trigonometric functions always be simplified using rewriting techniques?
4. What are the key trigonometric identities that help in solving such limits?
5. How can graphing functions provide insight into their limits?

Tip: Always verify the indeterminate form before applying L'Hôpital’s Rule to avoid unnecessary computation.