Answer
$\dfrac {1}{2}$
Work Step by Step
$\lim _{x\rightarrow 0}\dfrac {x}{\arctan 2x}=\dfrac {\dfrac {d}{dx}\left( x\right) }{\dfrac {d}{dx}\left( \arctan 2x\right) }=\dfrac {1}{\dfrac {2}{1+\left( 2x\right) ^{2}}}=\dfrac {1+4x^{2}}{2}=\dfrac {1+0}{2}=\dfrac {1}{2}$
