## Problem Statement

Calculate the limit of the function:

$\lim _{x \rightarrow 0}\left(\frac{\arcsin x}{x}\right)^{\frac{2}{x+5}}$
## Solution

$\lim _{x \rightarrow 0}\left(\frac{\arcsin x}{x}\right)^{\frac{2}{x+5}}=\left(\lim _{x \rightarrow 0} \frac{\arcsin x}{x}\right)^{\lim _{x \rightarrow 0} \frac{2}{x+5}}=$ $=\left(\lim _{x \rightarrow 0} \frac{\arcsin x}{x}\right)^{\frac{2}{0+5}}=\left(\lim _{x \rightarrow 0} \frac{\arcsin x}{x}\right)^{\frac{2}{5}}=$

Using the substitution of equivalent infinitesimals:

$\arcsin x \sim x$, as $x \rightarrow 0$

We get:

$=\left(\lim _{x \rightarrow 0} \frac{x}{x}\right)^{\frac{2}{5}}=\left(\lim _{x \rightarrow 0} 1\right)^{\frac{2}{5}}=1^{\frac{2}{5}}=1$

## Problem Kuznetsov Limits 18-23