https://file.aiquickdraw.com/mathbot/file/1728314615543zj5cmga0.jpg

The problem presented asks for the limit of \( \sec(x) \) as \( x \to 0 \), and includes a graph to support understanding. Here's the breakdown:

### Problem:
We need to evaluate:
\[
\lim_{x \to 0} \sec(x)
\]

### Step-by-step solution:
1. The function \( \sec(x) \) is defined as:
   \[
   \sec(x) = \frac{1}{\cos(x)}
   \]
2. As \( x \to 0 \), we know that \( \cos(0) = 1 \). Therefore:
   \[
   \sec(0) = \frac{1}{1} = 1
   \]
3. Since \( \sec(x) \) is continuous near \( x = 0 \), the limit as \( x \) approaches 0 from both sides exists and is equal to 1.

Thus, we conclude that:
\[
\lim_{x \to 0} \sec(x) = 1
\]

### Graph interpretation:
The graph in the image shows a curve resembling \( y = \sec(x) \), with vertical asymptotes at \( x = \pm \frac{\pi}{2} \), indicating that the function becomes undefined at those points. At \( x = 0 \), the value of the function is 1, as expected.

Would you like more detailed explanations or have any questions?

### Related questions:
1. What happens to \( \sec(x) \) as \( x \to \frac{\pi}{2} \)?
2. How does \( \sec(x) \) behave for values of \( x \) near \( -\frac{\pi}{2} \)?
3. What is the limit \( \lim_{x \to \pi} \sec(x) \)?
4. Can the secant function ever be negative? If so, when?
5. How is the secant function related to trigonometric identities?

### Tip:
The secant function has vertical asymptotes where the cosine function equals zero, which occurs at \( \frac{\pi}{2} + n\pi \), where \( n \) is an integer.

Explore how to find the limit of the secant function as x approaches 0. This problem focuses on key trigonometric concepts and limit theorems, demonstrating that the limit of sec(x) as x approaches 0 is 1. Suitable for high school students in Grades 11-12.

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