To evaluate the limit \(\lim_{{x \to 0}} |x| \sin(x)\), we can analyze it by considering the properties of the absolute value and the sine function.

### Step 1: Consider the behavior of \(|x|\) and \(\sin(x)\)
- The function \(|x|\) is the absolute value of \(x\). It behaves as \(x\) when \(x\) is positive and as \(-x\) when \(x\) is negative.
- The sine function \(\sin(x)\) is bounded between \(-1\) and \(1\) for all \(x\).

### Step 2: Establish bounds using the Squeeze Theorem
We can bound the product \(|x| \sin(x)\) between \(-|x|\) and \(|x|\):

\[
-|x| \leq |x| \sin(x) \leq |x|
\]

### Step 3: Apply the Squeeze Theorem
As \(x \to 0\), both \(-|x|\) and \(|x|\) approach 0. Since \(|x| \sin(x)\) is squeezed between these two functions, it must also approach 0.

Thus, by the Squeeze Theorem:

\[
\lim_{{x \to 0}} |x| \sin(x) = 0
\]

The final answer is:

\[
\lim_{{x \to 0}} |x| \sin(x) = 0
\]

Would you like any more detailed explanation on this, or do you have further questions?

Here are 5 related questions for further practice:
1. What is \(\lim_{{x \to 0}} x \sin\left(\frac{1}{x}\right)\)?
2. How would you evaluate \(\lim_{{x \to 0}} x^2 \sin\left(\frac{1}{x}\right)\)?
3. Can you find \(\lim_{{x \to 0}} |x| \cos(x)\)?
4. What is the limit \(\lim_{{x \to 0}} x^2 \sin(x)\)?
5. How do you apply the Squeeze Theorem to evaluate \(\lim_{{x \to 0}} \frac{\sin(x)}{x}\)?

**Tip:** The Squeeze Theorem is particularly useful when you have a product of a function that approaches 0 and a bounded function like sine or cosine.

Learn how to evaluate the limit of |x|sin(x) as x approaches 0 with a detailed step-by-step solution. This problem involves understanding limits, absolute values, the behavior of sine function near zero, and application of the Squeeze Theorem. Suitable for advanced math students.

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