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To solve the limit

\[
\lim_{x \to 0} x^4 \cos\left(\frac{1}{x^2}\right) = 0
\]

using the **Squeeze Theorem**, we need to apply the theorem based on how the cosine function behaves. The Squeeze Theorem states that if \( f(x) \leq g(x) \leq h(x) \) for all \( x \) in some interval, and if \( \lim_{x \to a} f(x) = \lim_{x \to a} h(x) = L \), then \( \lim_{x \to a} g(x) = L \).

### Step-by-Step Solution:

1. **Bound the cosine function**: 
   We know that for all \( x \), \( \cos(\theta) \) is always between -1 and 1, regardless of the value of \( \theta \). Thus, we have:
   \[
   -1 \leq \cos\left(\frac{1}{x^2}\right) \leq 1
   \]
   
2. **Multiply by \( x^4 \)**:
   Multiply the entire inequality by \( x^4 \) (note that \( x^4 \geq 0 \) for all real \( x \)) to get:
   \[
   -x^4 \leq x^4 \cos\left(\frac{1}{x^2}\right) \leq x^4
   \]
   
3. **Take the limit as \( x \to 0 \)**:
   Now, we take the limit of the outer bounds as \( x \to 0 \):
   \[
   \lim_{x \to 0} (-x^4) = 0 \quad \text{and} \quad \lim_{x \to 0} x^4 = 0
   \]
   
   Therefore, by the Squeeze Theorem:
   \[
   \lim_{x \to 0} x^4 \cos\left(\frac{1}{x^2}\right) = 0
   \]

### Conclusion:
By the Squeeze Theorem, we have shown that the limit of \( x^4 \cos\left(\frac{1}{x^2}\right) \) as \( x \to 0 \) is indeed 0.

Would you like further details or explanations?

Here are 5 related questions:
1. How is the Squeeze Theorem applied to more complex functions?
2. What happens to the limit if we replace \( x^4 \) with \( x^2 \) in the given expression?
3. Can you use the Squeeze Theorem when dealing with oscillating functions like sine and cosine?
4. How do we know that multiplying by \( x^4 \) preserves the bounds in this problem?
5. What other methods (besides the Squeeze Theorem) could be used to evaluate this limit?

**Tip**: When dealing with trigonometric functions in limits, bounding the function between known values is often a useful strategy for applying the Squeeze Theorem.

Use the Squeeze Theorem to show that lim (x→0) x^4 cos(1/x^2) = 0

Learn how to apply the Squeeze Theorem to evaluate the limit of x^4 cos(1/x^2) as x approaches 0. This problem involves key calculus concepts, including limits and trigonometric functions, suitable for college-level students.

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