I came across an interesting limit I could not solve:

$$
\lim\_{x \to 0^{+}}\left[\arcsin\left(x\right)\right]^{\tan\left(x\right)}
$$

Given we have not proven l'Hôpital's rule yet, I have to solve it without it. Also, I would rather not use advanced methods such as the taylor series (which yield $x^x$ here).

Squeeze theorem does not (easily?) really help here, nor does the exponent function as far as I see it:

$$
\lim\_{x\rightarrow 0+}(\arcsin x)^{\tan\,x} = 
\lim\_{x\rightarrow 0+} e^{{\tan(x)}\ln(\arcsin x)}
$$
Here again the exponent is an undefined term $(0 \cdot +\infty)$. Unlike all limits I practiced on however, this logarithm does not tend to $1$, so I don't really see how it cancels out.

Is there an easy solution I am missing?