Empty point or asymptote

Question

I've encountered the two functions: $$f(x) = \frac{1}{\ln x -1}$$ $$g(x) = \ln(8x-x^2)$$ I know that in $x=0$ both functions are undefined, but I can't really understand why in $ g(x)$ there is an asymptote in $ x = 0$ while in $f(x)$ there's an empty point in $x = 0$. Will be happy to an explaination, thanks in advance :)

Answer 1

Notice that as $x\to0^+$, we have

$$\lim_{x\to0}\frac1{\ln(x)-1}=0$$

Since $\ln(x)$ gets infinitely big, so the fraction gets infinitely small. Thus, that point is not an asymptote, but just an empty point.

It does, however, have asymptotes at $x=e$...