The function you've provided is:

\[
f(x) = 1 - \frac{\cos(x)}{x^2}
\]

Let's go through what this function represents and analyze its behavior:

1. **For \(x = 0\)**: The function is undefined because \( \frac{\cos(x)}{x^2} \) involves division by zero. To analyze this case, we might want to find the limit as \(x \to 0\) using techniques like L'Hôpital's Rule or series expansion.

2. **For \(x \neq 0\)**: The function behaves normally, as long as \(x\) is not zero.

Would you like a deeper analysis, such as finding limits, derivatives, or plotting the function? Also, are you interested in understanding this function’s behavior near zero?

Here are five related questions to explore:
1. What is the limit of \( f(x) \) as \(x \to 0\)?
2. What is the derivative of \( f(x) \)?
3. How does the function behave as \( x \to \infty \)?
4. Is the function continuous everywhere except at \(x = 0\)?
5. Can the function be expanded as a Taylor series around \(x = 0\)?

**Tip**: When faced with indeterminate forms at a point, using L'Hôpital's Rule or a Taylor expansion can provide insight into the function's behavior near that point.

Explore the function f(x) = 1 - cos(x)/x² with a detailed analysis of its behavior near x = 0, including the use of L'Hôpital's Rule and Taylor Series expansion. Learn about key calculus concepts such as limits, indeterminate forms, and series expansion. Suitable for advanced high school students and college level.

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