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Question Number 193248 by mnjuly1970 last updated on 08/Jun/23

$$$$$$\mathrm{L}={\lim }_{x\to 0}\frac{\sin \left(x\right)-\arcsin \left(x\right)}{\tan \left(x\right)-\arctan \left(x\right)}=?$$$$$$$$$$

Answered by MM42 last updated on 08/Jun/23

$$sinx-{sin}^{-1}x\sim -\frac{1}{3}{x}^3\mathrm{\&}tanx-{tan}^{-1}x\sim \frac{2}{3}{x}^3$$$$\Rightarrow L={lim}_{x\to 0}\frac{-\frac{1}{3}{x}^3}{\frac{2}{3}{x}^3}=-\frac{1}{2}$$

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