To compute $\lim_{x\to 0^+} x \log(x)$ one usually resorts to Hôpital's theorem which is often looked down by many in favour of Taylor's theorem which is a more flexible tool with many generalisations and applications.
However, in this specific case, since $\log(x)$ is not defined at zero we cannot consider the Taylor expansion at that point.
Is there a way to compute the above limit without Hôpital and using instead Taylor's expansion?
Another way (not exactly Taylor but essentially the same idea) would be to use asymptotic expansions of $\log(x)$ for $x\to 0$, these make sense even if $0$ is not in the domain of $\log$.
However I do not know of any asymptotic expansion that might be helpful.