$f_{1}=\frac{v}{2 l}$

The frequency of an organ pipe with length $(l+x)$

$f_{2}=\frac{v}{2(l+x)}$

where $v$ is the speed of sound in air

Then, for the beat frequency heard,

$f_{1}-f_{2}=\frac{v}{2 l}-\frac{v}{2(l+x)}$

$\Rightarrow f_{1}-f_{2}=\frac{v}{2 l}\left[1-\left(1+\frac{x}{l}\right)^{-1}\right]$

$\Rightarrow f_{1}-f_{2}=\frac{v}{2 l}\left[1-1+\frac{x}{l}\right]=\frac{v x}{2 l^{2}}$