Try out some values of $x$ .

Find the largest positive real number $x$ such that

$\frac{2}{x} = \frac{1}{\lfloor x \rfloor} + \frac{1}{\lfloor 2x \rfloor}$

where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ .

Give your answer to 3 decimal places.