give me an example where we need to use cos and squeeze theroem outline all steps and dont leave a single step 

Sure! I'll give you a detailed example where we need to use the **Squeeze Theorem** and the **cosine function**.

### Problem Example:
Find \(\lim_{x \to 0} x^2 \cos \left( \frac{1}{x} \right)\).

### Step-by-Step Solution:

1. **State the limit expression**:
   We are tasked with finding:
   \[
   \lim_{x \to 0} x^2 \cos \left( \frac{1}{x} \right)
   \]

2. **Recognize the form**:
   Notice that as \(x \to 0\), the term \(\cos \left( \frac{1}{x} \right)\) oscillates between \(-1\) and \(1\) infinitely fast. However, the function \(x^2\) approaches \(0\), and the behavior of \(x^2 \cos \left( \frac{1}{x} \right)\) is influenced by this product.

3. **Squeeze Theorem Setup**:
   To apply the Squeeze Theorem, we first need to bound the function \(x^2 \cos \left( \frac{1}{x} \right)\) between two simpler functions.

   Recall that for all real numbers \(t\),
   \[
   -1 \leq \cos(t) \leq 1
   \]
   So for any \(x \neq 0\), we have:
   \[
   -1 \leq \cos \left( \frac{1}{x} \right) \leq 1
   \]

4. **Multiply by \(x^2\)**:
   Multiply the entire inequality by \(x^2\) (which is non-negative for all \(x\)) to get bounds on \(x^2 \cos \left( \frac{1}{x} \right)\):
   \[
   -x^2 \leq x^2 \cos \left( \frac{1}{x} \right) \leq x^2
   \]

5. **Apply the Squeeze Theorem**:
   Now, observe the behavior of the bounding functions as \(x \to 0\). We know that:
   \[
   \lim_{x \to 0} -x^2 = 0 \quad \text{and} \quad \lim_{x \to 0} x^2 = 0
   \]
   Since both the lower bound \(-x^2\) and the upper bound \(x^2\) approach \(0\) as \(x \to 0\), by the **Squeeze Theorem**, we can conclude that:
   \[
   \lim_{x \to 0} x^2 \cos \left( \frac{1}{x} \right) = 0
   \]

### Final Answer:
\[
\lim_{x \to 0} x^2 \cos \left( \frac{1}{x} \right) = 0
\]

### Summary of Steps:
1. Recognize the problem involves an oscillating function (\(\cos \left( \frac{1}{x} \right)\)) and that \(\cos(t)\) is bounded by \(-1\) and \(1\).
2. Multiply by \(x^2\) to set up inequalities.
3. Apply the Squeeze Theorem, showing both bounding functions

give me an example where we need to use cos and squeeze theorem outline all steps and dont leave a single step

Learn how to solve the limit of x^2 cos(1/x) as x approaches 0 using the Squeeze Theorem. This example covers limits, trigonometric functions, and step-by-step application of the Squeeze Theorem, suitable for advanced high school or undergraduate calculus students.

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