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 $0$.
 A: Using "anything" in "... multiply anything with $0$..." is an abuse of language; according to the context, that "anything" can only be a complex number. But $\log 0$ is not even defined; hence the arithmetic multiplication does not apply to $\log 0$. 
If one "multiplies" an undefined thing by $0$, then he is not "playing the game".
A: The function $$f(x)=x\log x$$ has a singularity at $x=0$, since $\log 0$ is undefined. However, this singularity is removable, since $\lim_{x\to0}f(x)$ exists. Therefore you can define a new function
$$\tilde f(x)=\begin{cases}
x\log x&x\ne 0\\
0&x=0
\end{cases}$$
This is what you looking for I guess. 
btw, there is other types of singularities, which are not removable.
A: 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).
A: 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?
A: 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.
