lim --> pi sobre 2                  pi sobre 2 -x/ cos(x)

You have given a limit problem, but the notation seems incomplete. From what I understand, you want to evaluate the following limit:

\[
\lim_{x \to \frac{\pi}{2}} \frac{\frac{\pi}{2} - x}{\cos(x)}
\]

### Step-by-Step Solution:

1. **Substitute \(x = \frac{\pi}{2}\):**
   \[
   \frac{\frac{\pi}{2} - x}{\cos(x)} = \frac{\frac{\pi}{2} - \frac{\pi}{2}}{\cos\left(\frac{\pi}{2}\right)} = \frac{0}{0}
   \]
   This is an indeterminate form \( \frac{0}{0} \), so we can apply L'Hôpital's Rule.

2. **Apply L'Hôpital's Rule:**
   L'Hôpital's Rule states that for limits of the form \(\frac{0}{0}\), we can find the limit by differentiating the numerator and denominator separately.
   
   Differentiating the numerator \(\frac{d}{dx} \left( \frac{\pi}{2} - x \right)\):
   \[
   \frac{d}{dx} \left( \frac{\pi}{2} - x \right) = -1
   \]
   
   Differentiating the denominator \(\frac{d}{dx} \left( \cos(x) \right)\):
   \[
   \frac{d}{dx} \left( \cos(x) \right) = -\sin(x)
   \]

   So, the limit becomes:
   \[
   \lim_{x \to \frac{\pi}{2}} \frac{-1}{-\sin(x)} = \lim_{x \to \frac{\pi}{2}} \frac{1}{\sin(x)}
   \]

3. **Evaluate the limit:**
   As \(x\) approaches \(\frac{\pi}{2}\), \(\sin(x)\) approaches 1. Therefore:
   \[
   \lim_{x \to \frac{\pi}{2}} \frac{1}{\sin(x)} = \frac{1}{1} = 1
   \]

### Final Answer:
\[
\lim_{x \to \frac{\pi}{2}} \frac{\frac{\pi}{2} - x}{\cos(x)} = 1
\]

Would you like more details or have any questions?

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Here are 5 related questions to explore:

1. What is L'Hôpital's Rule, and when can it be applied?
2. How do you differentiate trigonometric functions like \(\cos(x)\) and \(\sin(x)\)?
3. What other types of indeterminate forms exist, and how are they resolved?
4. How would the limit change if the numerator was \(\frac{\pi}{2} + x\) instead?
5. How can we solve limits that do not result in indeterminate forms?

**Tip:** When dealing with indeterminate forms, always check if L'Hôpital's Rule is applicable—it simplifies many complex limits.

This guide provides a step-by-step solution to evaluating the limit of (pi/2 - x) / cos(x) as x approaches pi/2, using L'Hôpital's Rule to resolve the indeterminate form. Includes differentiation of trigonometric functions and practical applications of limit rules. Suitable for high school seniors and college-level calculus students.

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