use the squeeze thereom to evaluate lim(x^2sin1/x) as x approaches 0

To evaluate the limit \( \lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right) \) using the **Squeeze Theorem**, follow these steps:

### Step 1: Identify bounds for \( \sin\left(\frac{1}{x}\right) \)
The sine function is always bounded between \(-1\) and \(1\), regardless of its argument:
\[
-1 \leq \sin\left(\frac{1}{x}\right) \leq 1
\]

### Step 2: Multiply by \( x^2 \)
Multiply the entire inequality by \( x^2 \). Since \( x^2 \geq 0 \), the inequality's direction remains the same:
\[
- x^2 \leq x^2 \sin\left(\frac{1}{x}\right) \leq x^2
\]

### Step 3: Take the limit of the bounds
Now, take the limit of the left and right expressions as \( x \to 0 \):
\[
\lim_{x \to 0} -x^2 = 0
\quad \text{and} \quad
\lim_{x \to 0} x^2 = 0
\]

### Step 4: Apply the Squeeze Theorem
By the **Squeeze Theorem**, since \( -x^2 \leq x^2 \sin\left(\frac{1}{x}\right) \leq x^2 \) and both bounds tend to \(0\) as \(x \to 0\), the middle expression must also tend to \(0\). Therefore:
\[
\lim_{x \to 0} x^2 \sin\left(\frac{1}{x}\right) = 0
\]

Thus, the limit is \( \boxed{0} \).

Would you like further details or have any questions?

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Here are 5 related questions for further exploration:
1. How does the Squeeze Theorem apply to other trigonometric functions?
2. What happens if we replace \( \sin\left(\frac{1}{x}\right) \) with \( \cos\left(\frac{1}{x}\right) \)?
3. Can we evaluate \( \lim_{x \to 0} x^n \sin\left(\frac{1}{x}\right) \) for other powers \(n\)?
4. How can you apply the Squeeze Theorem to functions involving exponential limits?
5. How would the result change if we considered the limit as \( x \to \infty \)?

### Tip:
The Squeeze Theorem is especially useful when dealing with functions that oscillate but are bounded, like sine and cosine, where direct evaluation isn't possible.

use the squeeze theorem to evaluate lim(x^2sin1/x) as x approaches 0

Learn how to use the Squeeze Theorem to evaluate the limit lim(x^2sin(1/x)) as x approaches 0. This problem explores key calculus concepts like limits and trigonometric behavior, suitable for students in advanced high school or early college levels.

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