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

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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 ;).

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

## marked as duplicate by Simon S, Michael Albanese, Marconius, rogerl, Tim RaczkowskiNov 5 '15 at 1:26

• This looks remarkably similar to your previous question, and not in a good way. The sooner you change your approach to this site, the better your experience here will be. – user147263 Nov 4 '15 at 22:49
• I do not know how to start to solve this limit, because I never seen something like that – Marián Slovák Nov 4 '15 at 22:50
• @NormalHuman I had already flagged, asked my officemate to flag. – Silvia Ghinassi Nov 4 '15 at 22:55
• @SilviaGhinassi Thanks; it has some close votes now. – user147263 Nov 4 '15 at 22:56

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