To evaluate the given limit using L'Hôpital's Rule, let us proceed step by step.

The problem is:

$\lim_{x \to \frac{\pi}{2}} \left( \frac{\frac{2}{\cos x} - \sin x}{x - \frac{\pi}{2}} \right)$

Step 1: Check the indeterminate form

As $x \to \frac{\pi}{2}$, evaluate each term:

1. $\cos\left(\frac{\pi}{2}\right) = 0$, so $\frac{2}{\cos x} \to \infty$.
2. $\sin\left(\frac{\pi}{2}\right) = 1$, so $\frac{2}{\cos x} - \sin x \to \infty - 1 = \infty$.
3. The denominator $x - \frac{\pi}{2} \to 0$.

Thus, the limit is of the form $\frac{\infty}{0}$. This indicates a singularity, and L'Hôpital's Rule applies.

Step 2: Differentiate numerator and denominator

To apply L'Hôpital's Rule, differentiate the numerator $N(x)$ and denominator $D(x)$ with respect to $x$.

• Numerator: $N(x) = \frac{2}{\cos x} - \sin x$

Differentiate $N(x)$ term by term:

1. For $\frac{2}{\cos x}$, rewrite as $2 \sec x$. Its derivative is: $\frac{d}{dx} [2 \sec x] = 2 \sec x \tan x$
2. For $\sin x$, its derivative is: $\frac{d}{dx} [\sin x] = \cos x$

Thus, the derivative of the numerator is: $N'(x) = 2 \sec x \tan x - \cos x$

• Denominator: $D(x) = x - \frac{\pi}{2}$

The derivative of the denominator is: $D'(x) = 1$

Step 3: Apply L'Hôpital's Rule

Now, apply L'Hôpital's Rule:

$\lim_{x \to \frac{\pi}{2}} \frac{N(x)}{D(x)} = \lim_{x \to \frac{\pi}{2}} \frac{N'(x)}{D'(x)}$

Substitute $N'(x)$ and $D'(x)$:

$\lim_{x \to \frac{\pi}{2}} \frac{2 \sec x \tan x - \cos x}{1}$

Step 4: Evaluate the limit as $x \to \frac{\pi}{2}$

As $x \to \frac{\pi}{2}$:

1. $\sec\left(\frac{\pi}{2}\right) = \frac{1}{\cos\left(\frac{\pi}{2}\right)} \to \infty$
2. $\tan\left(\frac{\pi}{2}\right) = \frac{\sin\left(\frac{\pi}{2}\right)}{\cos\left(\frac{\pi}{2}\right)} \to \infty$
3. $\cos\left(\frac{\pi}{2}\right) = 0$

Thus, the term $2 \sec x \tan x \to \infty$, and the other term $-\cos x \to 0$.

Hence, the overall limit is infinite:

$\lim_{x \to \frac{\pi}{2}} \left( \frac{\frac{2}{\cos x} - \sin x}{x - \frac{\pi}{2}} \right) = \infty$

Final Answer:

The limit diverges to infinity.

Do you want me to explain any step in more detail or clarify any concept?

Here are 5 related questions to deepen your understanding:

1. What is L'Hôpital's Rule, and when can it be applied?
2. How do you differentiate trigonometric functions like $\sec x$ and $\tan x$?
3. What are common indeterminate forms in limits?
4. How do you handle singularities in limits?
5. How can you analyze the behavior of trigonometric functions near critical points like $x = \frac{\pi}{2}$?

Tip: Always verify that the original limit is in an indeterminate form before applying L'Hôpital's Rule.