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

Question

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

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

Answer 1

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 $0$.

Answer 2

Note that $f(x)=\ln^x(x)$ is defined when both the following conditions are fulfilled

• for existence of $\ln x$ we need $x>0$

• for existence of $a^x$ we need $\ln x>0 \implies x>1$

therefore the given function is defined for $x>1$ and therefore the limit for $x\to 0$ is meaningless.