Determine whether the function $\arcsin \left(\sqrt{1-2\log _e^2\left(x\right)}\right)$ is continuous or not

I don't know how to study the continuity of this function$$\arcsin \left(\sqrt{1-2\log _e^2\left(x\right)}\right)$$ Do I have to take the first and last value of its domain and pass them to limits or how is correct approach? Thanks for any response.

• First find the domain of this function, then prove that $\lim_{n\to\infty} f(x_n) = x$ for all sequences $x_n$ such that all $x_n$ are in the domain and with $x_n \to x$. – flawr Jun 30 '16 at 9:59
• Or, use the fact that $\sin$ is continuous, and the fact that $\sqrt{1-2\ln^2(x)}$ is also continuous. – Kenny Lau Jun 30 '16 at 10:04
• @flawr So all I have to do is compute the limit of $f(x)$ when $x \rightarrow \infty$ – T4yl0r Jun 30 '16 at 10:06
• @Kenny Lau Why is $\sqrt{1-2ln^2{x}}$ continuous? – T4yl0r Jun 30 '16 at 10:08
• @T4yl0r No, you have to do it for all the converging sequences in the domain. Just recall the definition of continuity. – flawr Jun 30 '16 at 10:17