Limit evaluate $ \lim\limits_{x \to - \infty } {{|\arcsin ({2 \over x})|} \over {\arctan ({5 \over x})}} $

Question

Please help me with this limit without using L'Hôpital's rule. I would by happy if you use simple solving. Thank you as much as I can ;).

$ \lim\limits_{x \to - \infty } {{|\arcsin ({2 \over x})|} \over {\arctan ({5 \over x})}} $

Answer 1

Hint

$$\frac{\arcsin(u)}{\arctan(v)}=\frac{\arcsin(u)}{u}\cdot \frac{u}{v}\cdot \frac{v}{\arctan(v)}.$$ Therefore, if $u(x),v(x)\underset{x\to a}{\longrightarrow} 0$, $$\lim_{x\to a}\frac{\arcsin(u(x))}{\arctan(v(x))}=\lim_{x\to a}\frac{u(x)}{v(x)}.$$

Answer 2

All you need to know are

(1) $|x| > 0$ if $x \ne 0$.

(2) $\lim_{x \to 0} \frac{f(x)}{x} = 1$ for all the functions and their inverses mentioned in this problem.