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 $0$.

It is true that $x\ln x$ is undefined for $x=0$ but that doesn't matter when we ask $\lim_{x\to 0^+} x\ln x$ because we are only asking what the value is approching arbitrarily close. 
However the whole "$0$ times anything equals zero" means "$0$ times any element within the ring is $0$". So does $\ln 0$ exist in reals? no it doesn't, it has no value, it is meaningless and hence it cannot be multiplied by $0$. It is like asking what is $0\cdot red$, red is not a number so how does multiplication work even then?
