Given any function, how can I find its vertical asymptote? I know that for rationals I can do this by letting the denominator equal to 0. But how about a function like:
$\ln(1-\ln(x))$? You can find the horizontal asymptote of any function by finding the limit of the function as it approaches positive and negative infinity. Is there a generalized algorithm like that for finding the vertical asymptote as well, even when the function is not a rational?
Best Answer
You simply pick off values of $x$ that are not defined in an orderly manner. For example, in the case of $\ln(1-\ln(x))$, look to the inner nest. By a property of logarithms, $\ln(x)$ is undefined for $x \leq 0$. There is a vertical asymptote at $x=0$, and the function is not defined for $x \leq 0$. Take the next nest, $$1-\ln(x) > 0 \Rightarrow \ln(x) <1 \Rightarrow e^{\ln(x)} < e^1 \Rightarrow x < e.$$ So now $x=e$ is another vertical asymptote, and $x<e$.
That is about it. Just collect exclusions for $x$ in an orderly manner. When an exclusion is a boundary as is the case here, it is generally going to be an asymptote.