## Problem Statement

Calculate the limit of the function:

$\lim _{x \rightarrow 0} \frac{1-\cos x}{\left(e^{3 x}-1\right)^{2}}$
## Solution

Let's use the substitution of equivalent infinitesimals:

$1-\cos x \sim \frac{x^{2}}{2}$, as $x \rightarrow 0$
$e^{3 x}-1 \sim 3 x$, as $x \rightarrow 0(3 x \rightarrow 0)$

We get:

$\lim _{x \rightarrow 0} \frac{1-\cos x}{\left(e^{3 x}-1\right)^{2}}=\left\{\frac{0}{0}\right\}=\lim _{x \rightarrow 0} \frac{\frac{x^{2}}{2}}{(3 x)^{2}}=$

$=\lim _{x \rightarrow 0} \frac{1}{2 \cdot 3^{2}}=\frac{1}{18}$

## Problem Kuznetsov Limits 11-24(2)