 Solution

The real logarithm restricts the approach to $x\to 0^{+}$. The Taylor expansion $\sin x-x=-x^3/6+O(x^5)$ givesThe positive first term dominates, since $x^2/|\ln x|\to 0$. Thus the requested extended limit is $\boxed{+\infty }$. There is no two-sided real limit across zero because the real logarithm is undefined for $x<0$.