Limits of inverse trigonometric fucnctions

Question

I was solving for $x$ if

$$\lim_{n \rightarrow\infty}(\sin^{-1}(x))^n = 0$$

and got the answer as $x$ is all values except $0$ and $\{\pi/2+n\pi\}$ for all integers $n$. Am I right ?

Answer 1

One has that $$\lim_{n\rightarrow\infty}|x|^n=\begin{cases}0, \quad \text{if } |x|<1 \\ 1, \quad \text{if }|x|=1 \\ +\infty, \quad \text{if } |x|>1\end{cases},$$ And in particular the first holds without absolute value.

Thus you need to find the inverse image of $(-1,1)$ under arcsin, i.e. $$\{x\in \mathbb{R}\mid |\arcsin(x)|<1\}\iff \{x\in \mathbb{R}\mid |x|<\sin(1)\}=(-\sin(1),\sin(1))$$