hey obtain the limit for the function (secx - tanx) as x approaches pi/2

To find the limit of the function \( \sec(x) - \tan(x) \) as \( x \) approaches \( \frac{\pi}{2} \), we can proceed step by step.

The function is:

\[
f(x) = \sec(x) - \tan(x)
\]

where:

- \( \sec(x) = \frac{1}{\cos(x)} \)
- \( \tan(x) = \frac{\sin(x)}{\cos(x)} \)

We now evaluate the limit:

\[
\lim_{x \to \frac{\pi}{2}} \left( \sec(x) - \tan(x) \right)
\]

### Step 1: Behavior of \( \sec(x) \) as \( x \to \frac{\pi}{2} \)

As \( x \to \frac{\pi}{2} \), \( \cos(x) \to 0 \). Thus, \( \sec(x) = \frac{1}{\cos(x)} \) tends to infinity because division by a very small number leads to a large value.

\[
\lim_{x \to \frac{\pi}{2}} \sec(x) = \infty
\]

### Step 2: Behavior of \( \tan(x) \) as \( x \to \frac{\pi}{2} \)

Similarly, as \( x \to \frac{\pi}{2} \), \( \tan(x) = \frac{\sin(x)}{\cos(x)} \) also tends to infinity because \( \sin(x) \) approaches 1 and \( \cos(x) \to 0 \).

\[
\lim_{x \to \frac{\pi}{2}} \tan(x) = \infty
\]

### Step 3: Difference \( \sec(x) - \tan(x) \) as \( x \to \frac{\pi}{2} \)

Now let's analyze the expression \( \sec(x) - \tan(x) \):

\[
\sec(x) - \tan(x) = \frac{1}{\cos(x)} - \frac{\sin(x)}{\cos(x)} = \frac{1 - \sin(x)}{\cos(x)}
\]

As \( x \to \frac{\pi}{2} \), \( \sin(x) \to 1 \) and \( \cos(x) \to 0 \). The numerator \( 1 - \sin(x) \) approaches 0, and the denominator approaches 0 as well.

Thus, the limit becomes:

\[
\lim_{x \to \frac{\pi}{2}} \frac{1 - \sin(x)}{\cos(x)} = 0
\]

### Conclusion:

Therefore, the limit of \( \sec(x) - \tan(x) \) as \( x \to \frac{\pi}{2} \) is:

\[
\boxed{0}
\]

Would you like further details on any of the steps or clarifications?

Here are 5 related questions to explore this topic further:
1. What happens if we take the limit of \( \sec(x) \) and \( \tan(x) \) individually as \( x \) approaches \( \pi/2 \) from the left and right?
2. How do we calculate limits when both numerator and denominator approach 0, like in \( \frac{0}{0} \) forms?
3. How can we find the limit of \( \cot(x) \) as \( x \to 0 \)?
4. What are some other trigonometric functions that exhibit interesting behavior near specific points like \( \frac{\pi}{2} \)?
5. How do we apply L'Hopital's Rule to functions that result in indeterminate forms?

**Tip**: Always check if the function you're dealing with involves any vertical asymptotes or points of discontinuity before directly applying limit rules.

Obtain the limit for the function (secx - tanx) as x approaches pi/2.

Explore how to find the limit of the trigonometric function sec(x) - tan(x) as x approaches pi/2. This solution involves key trigonometric concepts, including secant and tangent functions, limits, and indeterminate forms. Suitable for high school students in Grades 11-12.

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