Determine which of the following is a correct step to help show lim(x->0) x^2sin (3/x)=0 by using the squeeze theorem 

To determine which step helps to show \(\lim_{x \to 0} x^2 \sin\left(\frac{3}{x}\right) = 0\) using the **squeeze theorem**, let's break down the problem.

### Step 1: Understanding the Behavior of \(\sin\left(\frac{3}{x}\right)\)
The function \(\sin\left(\frac{3}{x}\right)\) oscillates between \(-1\) and \(1\) for all values of \(x\). This gives us the inequality:
\[
-1 \leq \sin\left(\frac{3}{x}\right) \leq 1
\]

### Step 2: Multiplying the inequality by \(x^2\)
Since \(x^2 \geq 0\) for all \(x\), multiplying all parts of the inequality by \(x^2\) gives:
\[
- x^2 \leq x^2 \sin\left(\frac{3}{x}\right) \leq x^2
\]

### Step 3: Applying the Squeeze Theorem
Now we take the limit of all parts of this inequality as \(x \to 0\):
\[
\lim_{x \to 0} -x^2 = 0 \quad \text{and} \quad \lim_{x \to 0} x^2 = 0
\]

Since both the left-hand side and the right-hand side limits are 0, by the **squeeze theorem**, we conclude:
\[
\lim_{x \to 0} x^2 \sin\left(\frac{3}{x}\right) = 0
\]

### Correct Step
The correct step is recognizing that \(-1 \leq \sin\left(\frac{3}{x}\right) \leq 1\) and multiplying it by \(x^2\) to squeeze the original function \(x^2 \sin\left(\frac{3}{x}\right)\).

Would you like more details or have any questions?

Here are some related questions to expand on this concept:
1. How does the oscillating nature of \(\sin(x)\) affect limit problems like this?
2. What is the formal definition of the squeeze theorem?
3. Why can't we directly compute the limit of \(\sin\left(\frac{3}{x}\right)\) as \(x \to 0\)?
4. Are there other trigonometric functions for which the squeeze theorem is useful in limits?
5. How does the behavior of \(x^2\) help in making the limit go to 0?

**Tip:** The squeeze theorem is especially useful for limits involving oscillating functions like sine or cosine.

Determine which of the following is a correct step to help show lim(x->0) x^2sin(3/x)=0 by using the squeeze theorem.

Explore how to apply the squeeze theorem to determine lim(x->0) x^2 sin(3/x) = 0. This step-by-step guide covers key concepts in calculus, including limits, trigonometric functions, and the squeeze theorem. Ideal for students in Grades 11-12 or early college-level calculus.

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