Peculiar similarity

Question

I am an experimental physicist and have a math question. There are two functions, one being

$-1/\log(x)$

the other

$\sqrt{x}/(1-x)$

which are remarkably similar in the domain ]0,1[.

Does anybody have an idea why?

Answer 1

I just figured it out myself. Develop $\log(x)$ around 1, then write out $-\frac{1}{\log(x)}$, factor out (1-x) and then write $\sqrt(x)$ as $1/1/\sqrt(x)$ and develop $1/\sqrt(x)$ around 1, then compare.

Answer 2

The approximation $\ln x\approx x-1$ shows these functions are similar when $x$ is close to $1$. When $x$ is small and positive, so are both functions. But they look less similar if you zoom to say $0\le x\le 0.1$, and even less similar if you zoom to $0\le x\le 0.001$. Indeed,$$\lim_{x\to0^+}\frac{\sqrt{x}/(1-x)}{-1/\ln x}=\lim_{x\to0^+}\frac{-2\sqrt{x}\ln\sqrt{x}}{1-x}=-2\lim_{y\to^+}y\ln y=0.$$

Answer 3

The series development around $x=1$ gives: $$ \eqalign{ & {{\sqrt {\left( {1 + x} \right)} } \over {\left( {1 - \left( {1 + x} \right)} \right)}} = - {{\sqrt {\left( {1 + x} \right)} } \over x} = - \left( {{1 \over x} + {1 \over 2} - {x \over 8} + {{x^{\,2} } \over {16}} + O\left( {x^{\,3} } \right)} \right) \cr & - {1 \over {\ln \left( {1 + x} \right)}} = - \left( {{1 \over x} + {1 \over 2} - {x \over {12}} + {{x^{\,2} } \over {24}} + O\left( {x^{\,3} } \right)} \right) \cr} $$