Compute $\lim_{x\to0} \ln^x(x)$ 
$$\lim_{x\to0} \ln^x(x)$$

I know I can use L'Hospital and get an answer equal to one.
But how can he ask a question like this even if the function is not continuous from $[0, 1]$. It has only discrete solutions not a graph

The given function doesn't have discrete values if you plot it on its natural domain.
Speaking about the natural domain of the function, since you're using a real exponent, which is typically defined as $a^x=e^{a\ln x}$, the natural domain of your function is $[1,+\infty)$. Hence you can't talk about a limit around $0$.
