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

Question

The other day, while I was accessing a Russian math enthusiast web site, I came across this captcha:

Can you help me make sense of it, and enter that site?

Answer 1

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

Thus,

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

And

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

Answer 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:

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

Answer 3

Just for the sake of accuracy, I want to add that other answers are correct if it is assumed that the limit in question is one-sided. In other words, they show:

$$\lim_{x\to{0^+}}\ ln\left(2+\sqrt{\arctan(x)\sin(1/x)}\right)=\ln(2)$$

(notice the plus sign in $x\to{0^+}$)

However (see the comments to the question and to the answers), it is clear that the two-sided limit does not exist.