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The limit expression you provided is:

\[
\lim_{{x \to +\infty}} x^2 \left( 1 - \cos \left( \frac{1}{x} \right) \right)
\]

### Steps to solve the limit:

1. **Analyze the behavior of \(\cos\left( \frac{1}{x} \right)\) as \(x \to \infty\):**

   As \(x \to \infty\), \(\frac{1}{x} \to 0\). We know that:
   \[
   \cos(0) = 1
   \]
   Therefore, \(\cos\left( \frac{1}{x} \right) \to 1\) as \(x \to \infty\).

2. **Use the small angle approximation:**

   For very small angles, we can use the approximation:
   \[
   1 - \cos(\theta) \approx \frac{\theta^2}{2} \quad \text{when} \quad \theta \to 0
   \]
   In this case, \(\theta = \frac{1}{x}\), so:
   \[
   1 - \cos\left( \frac{1}{x} \right) \approx \frac{\left(\frac{1}{x}\right)^2}{2} = \frac{1}{2x^2}
   \]

3. **Substitute the approximation into the limit expression:**
   \[
   \lim_{{x \to +\infty}} x^2 \left( 1 - \cos\left( \frac{1}{x} \right) \right) \approx \lim_{{x \to +\infty}} x^2 \cdot \frac{1}{2x^2}
   \]

4. **Simplify the expression:**
   \[
   \lim_{{x \to +\infty}} \frac{1}{2} = \frac{1}{2}
   \]

### Final Answer:
\[
\lim_{{x \to +\infty}} x^2 \left( 1 - \cos \left( \frac{1}{x} \right) \right) = \frac{1}{2}
\]

Would you like further clarification or details on this? 

Here are some related questions to explore:
1. How do small angle approximations work in other trigonometric limits?
2. What happens to the limit if we replace \( \cos\left(\frac{1}{x}\right) \) with \( \sin\left(\frac{1}{x}\right) \)?
3. Can L'Hopital's rule be applied to this limit? If so, how?
4. How would the solution change if the limit was as \( x \to 0 \) instead of \( x \to \infty \)?
5. What role does Taylor series expansion play in solving limits involving trigonometric functions?

**Tip:** For limits involving trigonometric functions as their argument tends to zero, small angle approximations or Taylor series expansions can often simplify calculations.

Calculate the limit as x approaches infinity for the expression x^2 (1 - cos(1/x)).

Explore the detailed calculation of the limit of x^2 (1 - cos(1/x)) as x approaches infinity using the small angle approximation. This problem illustrates key concepts in calculus, specifically limits and trigonometric functions.

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