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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.

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

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