What is the value of $ x(\log x)$ when $x=0$ and $x\not \to 0$? I know that  $ x(\log x)\to 0$ when  $x \to 0$, but some people say that since $\log x$ is not defined when  $x=0$, so the value of $ x(\log x)$ can also not be found when $x=0$.
But isn't it true that if you multiply anything with 0, the answer comes out to be 0? According to me the value of $x(\log x)$ at $x=0$ should be $0$.

The logarithmic function is not defined at $x=0$. If you draw the graph as below, you will see that $x=0$ or y-axis is an asymptote to $\log x$. 

Hence anything undefined multiplied by $0$ yields only one result: UNDEFINED.
