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To compute the limit

\[
\lim_{x \to 0^+} 2 \sin(x) \ln(x),
\]

we’ll examine the behavior of each part of the expression as \( x \to 0^+ \).

1. **Examine \(\sin(x)\):**
   - As \( x \to 0^+ \), \( \sin(x) \approx x \). So, \( \sin(x) \) approaches \( 0 \) from the positive side.

2. **Examine \(\ln(x)\):**
   - As \( x \to 0^+ \), \( \ln(x) \to -\infty \).

This means we are dealing with an indeterminate form of \( 0 \cdot (-\infty) \), which suggests rewriting it as a fraction for L'Hôpital's Rule.

### Rewriting the Expression

Rewrite \( 2 \sin(x) \ln(x) \) as:
\[
2 \sin(x) \ln(x) = \frac{2 \ln(x)}{\frac{1}{\sin(x)}}.
\]

Now, as \( x \to 0^+ \):
- \( \ln(x) \to -\infty \),
- \( \frac{1}{\sin(x)} \to \infty \).

So this is an \(\frac{-\infty}{\infty}\) form, which allows us to apply L'Hôpital's Rule.

### Applying L'Hôpital's Rule

Differentiate the numerator and the denominator with respect to \( x \):

1. **Numerator \( 2 \ln(x) \):**
   \[
   \frac{d}{dx}(2 \ln(x)) = \frac{2}{x}.
   \]

2. **Denominator \( \frac{1}{\sin(x)} \):**
   \[
   \frac{d}{dx} \left( \frac{1}{\sin(x)} \right) = -\frac{\cos(x)}{\sin^2(x)}.
   \]

So the limit becomes:
\[
\lim_{x \to 0^+} \frac{\frac{2}{x}}{-\frac{\cos(x)}{\sin^2(x)}} = \lim_{x \to 0^+} \frac{-2 \sin^2(x)}{x \cos(x)}.
\]

### Simplifying Further

Using \( \sin(x) \approx x \) as \( x \to 0 \), we get:
\[
\lim_{x \to 0^+} \frac{-2 x^2}{x \cdot 1} = \lim_{x \to 0^+} -2x = 0.
\]

### Conclusion
\[
\lim_{x \to 0^+} 2 \sin(x) \ln(x) = 0.
\]

Would you like more detailed steps or have any questions?

---

1. Why do we approximate \(\sin(x)\) by \(x\) as \(x \to 0\)?
2. What other types of indeterminate forms can we solve using L'Hôpital's Rule?
3. Can you apply L'Hôpital's Rule multiple times if needed?
4. Why is \(\ln(x) \to -\infty\) as \(x \to 0^+\)?
5. What is the significance of rewriting the limit in fraction form?

**Tip:** When using L'Hôpital's Rule, make sure each application leads closer to a solvable limit to avoid repetitive differentiation.

Compute the following limit using L'Hôpital's rule if appropriate. Use INF to denote ∞ and MINF to denote -∞. \( \lim_{x \to 0^+} 2 \sin(x) \ln(x) \)

Learn how to compute the limit \( \lim_{x \to 0^+} 2 \sin(x) \ln(x) \) using L'Hôpital's rule to address the indeterminate form. This problem demonstrates key calculus concepts such as limits, trigonometric approximations, and logarithmic functions. Suitable for undergraduate calculus students.

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