Does y=1/lnx has an asymptote at x=0?

Question

Does $y=1/\ln x$ graph has an asymptote at $x=0$? It seems like its approaching to $0$ but some high-school math books say it has an asymptote at that point, because $0$ is out of domain. The $x=0$ will never exist. Is this statement true? (Talking about Real numbers not complex)

Answer 1

$f(x)=\ln x$ has "one-sided" asymptote at $x=0$ while $f(x)=\frac{1}{\ln x}$ does not. Recall that $f(x)$ has a one-sided asymptote at $x=x_0$ if $$ \lim_{x\to x_0^{+}}|f(x)|=\infty\qquad\text{or}\qquad \lim_{x\to x_0^{-}}|f(x)|=\infty, $$ not matter whether $f(x_0)$ is defined. In the case $f(x)=\ln x$ indeed $\lim_{x\to 0^{+}}|\ln x|=\infty$ but $$ \lim_{x\to 0^+}\left|\frac{1}{\ln x}\right|=0\neq\infty $$