The limit of the given expression $\lim_{x \to 0} \frac{\sec x - 1}{\tan^2 x}$ is $\frac{1}{2}$.

Would you like a detailed explanation on how this limit was solved?

Here are 5 related questions to explore:

1. What is the relationship between $\sec(x)$ and $\tan(x)$?
2. How can we simplify limits involving trigonometric functions near zero?
3. What is L'Hopital's rule and when can it be applied to limits?
4. How can the small angle approximations for sine and cosine help in solving limits?
5. How can you find the limit of a rational trigonometric function using series expansion?

Tip: When dealing with trigonometric limits near zero, small-angle approximations like $\tan x \approx x$ and $\sec x \approx 1 + \frac{x^2}{2}$ are very useful.