Solving math captcha involving a limit and $\sin(1/x)$

$$|\arctan(x)\sin(1/x)|<|x|$$

Thus,

$$\lim_{x\to0}\arctan(x)\sin(1/x)=0$$

And

$$\lim_{x\to0}\ln\left(2+\sqrt{\arctan(x)\sin(1/x)}\right)=\ln(2)$$

Since $$\lim_{x \to 0} (\arctan x)=0 \tag1$$

and

$$\lim_{x \to 0} \sin \left(\frac 1x \right) \mbox{ is oscillatory, but} \sin \left(\frac 1x \right)\in [-1,1] \tag2$$

By $(1)$ and $(2)$ , $$\lim_{x \to 0} \left[(\arctan x) \cdot \sin \left(\frac 1x \right) \right]=0 $$ Therefore:

$$\lim_{x\to0}\ln\left(2+\sqrt{\arctan(x)\sin\left(\frac1x\right)}~\right)=\boxed{\ln(2)}$$