Past exam of the mathematical methods II course of the University of Oxford / 2022 / 1

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 $x<0$.

For $x>2$, writeSince $|\sin x|\le 1$, the denominator factor tends to one, and $\ln x/x\to 0$. Therefore the limit is $\boxed{0}$. The function is eventually negative, so it approaches zero from below. Oscillation of $\sin x$ does not obstruct this limit, because it is bounded while $x$ grows.