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The limit presented is:

\[
\lim_{x \to 0^+} 8 \sin(x) \ln(x)
\]

### Step-by-step solution:

1. **Simplifying the expression**:
   The given expression is \( 8 \sin(x) \ln(x) \). The \( 8 \) is a constant, so we focus on the behavior of \( \sin(x) \ln(x) \).

2. **Behavior of the terms as \( x \to 0^+ \)**:
   - \( \sin(x) \) behaves as \( x \) for small \( x \), since \( \sin(x) \approx x \) when \( x \to 0 \).
   - \( \ln(x) \) tends to \( -\infty \) as \( x \to 0^+ \).

   Therefore, the product \( \sin(x) \ln(x) \) becomes \( x \ln(x) \).

3. **Analyzing the product \( x \ln(x) \)**:
   We now focus on finding the limit:
   
   \[
   \lim_{x \to 0^+} x \ln(x)
   \]

   This is an indeterminate form of type \( 0 \times (-\infty) \), so we can rewrite it as a fraction to apply **L'Hopital's Rule**:

   \[
   x \ln(x) = \frac{\ln(x)}{1/x}
   \]

4. **Applying L'Hopital's Rule**:
   To resolve this indeterminate form \( \frac{-\infty}{\infty} \), we differentiate the numerator and the denominator:

   - Derivative of \( \ln(x) \) is \( \frac{1}{x} \),
   - Derivative of \( \frac{1}{x} \) is \( -\frac{1}{x^2} \).

   So we now have:

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

5. **Conclusion**:
   Since \( \lim_{x \to 0^+} x \ln(x) = 0 \), multiplying by 8 gives:

   \[
   \lim_{x \to 0^+} 8 \sin(x) \ln(x) = 0
   \]

Thus, the value of the limit is:

\[
\boxed{0}
\]

Would you like further details or have any questions?

### Related Questions:
1. How is L'Hopital's Rule applied to more complex indeterminate forms?
2. What happens to the limit if we replace \( \sin(x) \) with another function like \( \cos(x) \)?
3. How does the behavior of logarithmic functions near 0 impact limits?
4. Can this limit be solved using series expansion for \( \sin(x) \)?
5. What are other common cases where L'Hopital's Rule is used?

### Tip:
When dealing with limits involving logarithms, rewriting the expression in a fractional form often helps apply L'Hopital's Rule efficiently.

Compute the following limit using l'Hopital's rule if appropriate: lim(x→0+) 8sin(x)ln(x)

Learn how to compute the limit lim(x→0+) 8sin(x)ln(x) using L'Hopital's Rule. This step-by-step guide explores trigonometric and logarithmic function behavior and applies differentiation to resolve the indeterminate form.

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