Limit of $x \log x$ as $x$ tends to $0^+$ Why is the limit of $x \log x$ as $x$ tends to $0^+$, $0$?


*

*The limit of $x$ as $x$ tends to $0$ is $0$.

*The limit of $\log x$ as $x$ tends to $0^+$ is $-\infty$.

*The limit of products is the product of each limit, provided each limit exists.

*Therefore, the limit of $x \log x$ as $x$ tends to $0^+$ should be $0 \times (-\infty)$, which is undefined and not $0$.


By $x=e^{-y}$ with $y \to \infty$ we have
$$x \log x =-\frac y {e^y} \to 0$$
which can be easily proved by the definition of $e^y$ or by induction and extended to reals.
