How I can find the vertical asymptotic of functions like:
$$y = \frac{\ln^2(x) - 1 }{\ln(x)}$$
$y = \dfrac{2}{\ln(x)}.$ (I didn't understood why the function doesn't have a vertical asymptotic at $x = 0.$)
And what is the technical way and steps needed to find the vertical asymptotic of such functions(without placing numbers to see if the limit tends to infinity...)?
We need to look at the limit at the points at which $\ln x=0$, that is
$$\lim_{x\to 1^+}\frac{\ln^2 x - 1 }{\ln x}=-\infty$$
$$\lim_{x\to 1-}\frac{\ln^2 x - 1 }{\ln x}=\infty$$
and also
$$\lim_{x\to 0^+}\frac{\ln^2 x - 1 }{\ln x}=\lim_{x\to 0^+} \ln x -\frac1{\ln x}\to -\infty$$
but for you second example
$$\lim_{x\to 0^+}\frac{2}{\ln x}=0$$