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The problem requires finding the limit:

\[
\lim_{{x \to 0}} 2x \cdot \ln(x^2)
\]

To solve this, let's analyze the expression inside the limit:

### Step 1: Simplify the logarithmic term

The logarithm property \(\ln(a^b) = b \cdot \ln(a)\) can be applied:

\[
\ln(x^2) = 2\ln(x)
\]

So the limit becomes:

\[
\lim_{{x \to 0}} 2x \cdot 2\ln(x) = 4x \cdot \ln(x)
\]

### Step 2: Analyze the behavior of \(4x \cdot \ln(x)\) as \(x \to 0\)

As \(x\) approaches 0 from the positive side, \( \ln(x) \) approaches \(-\infty\). The product \( x \cdot \ln(x) \) tends towards 0 but needs a more detailed evaluation:

Let's rewrite \(x \ln(x)\) using L'Hôpital's Rule:

Consider \( f(x) = x \ln(x) \). We can express this as:

\[
\lim_{{x \to 0^+}} \frac{\ln(x)}{\frac{1}{x}}
\]

This is an indeterminate form \( \frac{-\infty}{\infty} \), so L'Hôpital's Rule can be applied:

1. Differentiate the numerator: \(\frac{d}{dx}[\ln(x)] = \frac{1}{x}\).
2. Differentiate the denominator: \(\frac{d}{dx}\left[\frac{1}{x}\right] = -\frac{1}{x^2}\).

Now, applying L'Hôpital's Rule:

\[
\lim_{{x \to 0^+}} \frac{\frac{1}{x}}{-\frac{1}{x^2}} = \lim_{{x \to 0^+}} -x = 0
\]

Thus:

\[
\lim_{{x \to 0}} 4x \ln(x) = 4 \cdot 0 = 0
\]

### Final Answer:
\[
\lim_{{x \to 0}} 2x \cdot \ln(x^2) = 0
\]

Would you like further details on this process, or do you have any additional questions?

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Here are five related questions that can help you explore the topic further:

1. What is L'Hôpital's Rule, and when is it applicable in limit problems?
2. How do logarithmic properties simplify complex expressions?
3. What is the significance of evaluating limits approaching infinity or zero?
4. Can you solve other limit problems involving logarithms or exponential functions?
5. How do indeterminate forms affect the process of evaluating limits?

**Tip:** Always check if the limit results in an indeterminate form like \(0/0\) or \(\infty/\infty\) before applying L'Hôpital's Rule.

Learn how to solve the limit problem \(\lim_{x \to 0} 2x \cdot \ln(x^2)\) step-by-step using logarithmic properties and L'Hôpital's Rule. This problem involves evaluating the limit as \(x\) approaches 0, incorporating advanced concepts in calculus and logarithms.

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