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 $0$.

Since $\ln(x)$ is not defined, $x\ln(x)$ is not defined. Difficult to multiply things that do not exist.
However, since the existing definitions do not cover the case $\ln(0)$, you could arbitrarily choose a value for $\ln(0)$: $0$, $1$, $e$, $1664$ or your mother's birthday... There is no problem doing that, but there is no interest since the properties that works for $\ln(x)$, when $x$ is positive, would no longer be true if you include $0$ in the domain of $\ln$: the function will no longer be continuous on its whole domain and the theorem $\ln(xy)=\ln(x)+\ln(y)$ is not true if one of the numbers $x$ or $y$ is equal to $0$.
The same apply to $0\ln(0)$. In that case, many people (for example, people working with entropy) defines $x\ln(x)$ as $0$ when $x=0$. They find it convenient since it makes the functions $x\ln(x)$ continuous on $[0,\infty)$ and it is all that matters for them (I am maybe exagerating here).
