Answer
$\frac{\pi}{3}$
Work Step by Step
The rational function inside the arcsine is continous for all $x\in \mathbb R$ so the limit is equal to:
$$\arccos(\lim_{x \to \infty}\frac{1+x^2}{1+2x^2})$$
$$\arccos(\lim_{x \to \infty}\frac{x^2(1+\frac{1}{x^2})}{x^2(2+\frac{1}{x^2})})$$
$$\arccos(\lim_{x \to \infty}\frac{1+\frac{1}{x^2}}{2+\frac{1}{x^2}})$$
$$\arccos(\frac{1+0}{2+0})=\arccos(\frac{1}{2})=\frac{\pi}{3}$$
