limit(x tends to 0)(x/tan-1(2x))

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$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ \mathrm{L}=\underset{\mathrm{x}\to 0}{\lim } \frac{\mathrm{x}}{{\tan }^{-1}\left(2\mathrm{x}\right)} \\ =\frac{1}{2}\underset{\mathrm{x}\to 0}{\lim } \frac{\left(2\mathrm{x}\right)}{{\tan }^{-1}\left(2\mathrm{x}\right)} \\ \mathrm{By} \mathrm{standard} \mathrm{form} \mathrm{we} \mathrm{have}: \\ \underset{\mathrm{x}\to \mathrm{a}}{\lim } \frac{\mathrm{f}\left(\mathrm{x}\right)}{{\tan }^{-1}\left[\mathrm{f}\left(\mathrm{x}\right)\right]}=1, \mathrm{where} \underset{\mathrm{x}\to \mathrm{a}}{\lim } \mathrm{f}\left(\mathrm{x}\right)=0 \\ \mathrm{Hence} \underset{\mathrm{x}\to 0}{\lim } \frac{\left(2\mathrm{x}\right)}{{\tan }^{-1}\left(2\mathrm{x}\right)}=1 \\ \Rightarrow \mathrm{L}=\frac{1}{2}\times 1=\frac{1}{2} \\ \Rightarrow \boxed{\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{lim}}\mathbf{ }\frac{\mathbf{x}}{{\mathbf{tan}}^{\mathbf{-}\mathbf{1}}\left(\mathbf{2}\mathbf{x}\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}} \end{gathered}$$

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