$$\lim _{x \rightarrow 0}\left[\frac{\sin (\sin x)}{\sin (\arctan x)}\right]^{\frac{1}{1-\cos x}}$$

$\sin x \rightarrow 0, \quad \arctan x \rightarrow 0, \quad \sin x \sim x$,
$\ln (x+1) \sim x$

\begin{aligned} & \lim _{x \rightarrow 0}\left[\frac{\sin (\sin x)}{\sin (\arctan x)}\right]^{\frac{1}{1-\cos x}} \\ & =\exp \lim _{x \rightarrow 0} \frac{1}{1-\cos x} \ln \left[\frac{\sin (\sin x)}{\sin (\arctan x)}\right] \\ & =\exp \lim _{x \rightarrow 0} \frac{1}{1-\cos x} \ln \left[\left(\frac{\sin (\sin x)}{\sin (\arctan x)}-1\right)+1\right] \\ & =\exp \lim _{x \rightarrow 0} \frac{1}{1-\cos x}\left[\frac{\sin (\sin x)}{\sin (\arctan x)}-1\right] \\ & =\exp \lim _{x \rightarrow 0} \frac{1}{1-\cos x}\left[\frac{\sin (\sin x)-\sin (\arctan x)}{\sin (\arctan x)}\right] \end{aligned}

$$\frac{\sin (\sin x)-\sin (\arctan x)}{\sin x-\arctan x}=\cos (\xi)$$

$(\sin x-\arctan x) \sim \frac{1}{6} x^3$
$\arctan x \sim x$

\begin{aligned} & \exp \lim _{x \rightarrow 0} \frac{1}{1-\cos x}\left[\frac{\sin (\sin x)-\sin (\arctan x)}{\sin (\arctan x)}\right] \\ & =\exp \lim _{x \rightarrow 0} \frac{1}{1-\cos x}\left(\frac{\sin x-\arctan x}{\arctan x}\right) \\ & =\exp \lim _{x \rightarrow 0} \frac{1}{\frac{1}{2} x^2} \times \frac{\frac{1}{6} x^3}{x} \\ & =\exp \frac{1}{3} \end{aligned}

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