\displaystyle\lim_{x\to \frac{\pi}{2}}\sec(x)=?

To solve the limit:

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

Recall that \(\sec(x) = \frac{1}{\cos(x)}\). Therefore, the limit becomes:

\[
\lim_{x \to \frac{\pi}{2}} \frac{1}{\cos(x)}
\]

Now, we know that:

\[
\cos\left(\frac{\pi}{2}\right) = 0
\]

Thus, we are trying to evaluate:

\[
\frac{1}{0}
\]

Since division by zero is undefined, the secant function approaches infinity as \(x\) approaches \(\frac{\pi}{2}\). More precisely:

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

Therefore, the limit is:

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

Would you like a more detailed breakdown or have any further questions? Here are some related questions to expand on this topic:

1. What is the behavior of \(\sec(x)\) as \(x\) approaches other points where \(\cos(x) = 0\)?
2. How does the secant function behave around the vertical asymptotes?
3. Can you find the limit of \(\sec(x)\) as \(x \to 0\)?
4. What is the limit of \(\csc(x)\) as \(x \to 0\)?
5. How do the sine and cosine functions behave as \(x\) approaches \(\frac{\pi}{2}\)?

**Tip**: When dealing with trigonometric limits involving secant, it's helpful to remember that \(\sec(x)\) is undefined wherever \(\cos(x) = 0\).

What is the limit of \(\sec(x)\) as \(x\) approaches \(\frac{\pi}{2}\)?

Understand how to solve the limit of the secant function as x approaches π/2. This problem involves trigonometric concepts, such as secant and cosine functions, and examines their behavior as the limit approaches a vertical asymptote. Perfect for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation