1. **Problem statement:** Calculate the limit $$\lim_{x \to 0^+} (\sin x \cdot \ln x)$$ where $x$ approaches $0$ from the right (since $\ln x$ is defined only for $x>0$). 2. **Recall the behavior of functions near 0:** - $\sin x \approx x$ when $x \to 0$. - $\ln x \to -\infty$ as $x \to 0^+$. 3. **Rewrite the expression using the approximation:** $$\sin x \cdot \ln x \approx x \cdot \ln x$$ 4. **Evaluate the limit of $x \ln x$ as $x \to 0^+$:** Use substitution or known limit: $$\lim_{x \to 0^+} x \ln x = 0$$ This is because $\ln x$ tends to $-\infty$ but $x$ tends to $0$ faster, making the product tend to $0$. 5. **Conclusion:** Therefore, $$\lim_{x \to 0^+} (\sin x \cdot \ln x) = 0$$ **Final answer:** $0$