The given problem asks you to evaluate the following limit:

$\lim_{x \to 0} x \sin \left( \frac{9}{x} \right)$

using the Squeeze Theorem. Additionally, it is given that:

$-|x| \leq x \sin\left( \frac{9}{x} \right) \leq |x| \quad \text{for all } x \neq 0.$

Let's proceed step by step to solve the problem.

Step 1: Understanding the Bounds

The expression given tells us that the function $x \sin\left( \frac{9}{x} \right)$ is squeezed between $-|x|$ and $|x|$:

$-|x| \leq x \sin\left( \frac{9}{x} \right) \leq |x| \quad \text{for all } x \neq 0.$

Step 2: Setting Up the Squeeze Theorem

The Squeeze Theorem states that if we have a function $f(x)$ that lies between two other functions $g(x)$ and $h(x)$, and if:

$\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L,$

then:

$\lim_{x \to a} f(x) = L.$

In our case:

• $f(x) = x \sin\left( \frac{9}{x} \right)$
• $g(x) = -|x|$
• $h(x) = |x|$

We have:

$-|x| \leq x \sin\left( \frac{9}{x} \right) \leq |x|$

Step 3: Taking the Limit of the Bound Functions

Let's evaluate the limits of $g(x)$ and $h(x)$ as $x \to 0$:

$\lim_{x \to 0} -|x| = 0, \quad \lim_{x \to 0} |x| = 0$

Since both $-|x|$ and $|x|$ approach $0$ as $x \to 0$, we can use the Squeeze Theorem.

Step 4: Applying the Squeeze Theorem

By the Squeeze Theorem, since:

$-|x| \leq x \sin\left( \frac{9}{x} \right) \leq |x| \quad \text{and} \quad \lim_{x \to 0} -|x| = \lim_{x \to 0} |x| = 0,$

it follows that:

$\lim_{x \to 0} x \sin\left( \frac{9}{x} \right) = 0$

Final Answer

$\lim_{x \to 0} x \sin\left( \frac{9}{x} \right) = 0$

Would you like me to provide more details or explain any part further?

Here are some related questions to deepen your understanding:

1. How does the Squeeze Theorem help in evaluating limits with oscillating functions?
2. What is the behavior of $\sin\left( \frac{9}{x} \right)$ as $x$ approaches 0?
3. Can you apply the Squeeze Theorem to $\lim_{x \to 0} x \cos\left( \frac{1}{x} \right)$? If so, what is the result?
4. What are the limitations of using the Squeeze Theorem?
5. How would you approach this limit without using the Squeeze Theorem?

Tip: The Squeeze Theorem is particularly useful when dealing with limits involving oscillatory functions, especially when the amplitude of oscillation becomes smaller as you approach the limit point.