## Problem Statement

Calculate the limit of the function:

$\lim _{x \rightarrow 0} \frac{1-\cos 10 x}{e^{-x^{2}}-1}$
## Solution

Let's use the substitution of equivalent infinitesimals:

$$
\begin{aligned}
& 1-\cos 10 x \sim \frac{(10 x)^{2}}{2}, \text{ as } x \rightarrow 0(10 x \rightarrow 0) \\
& \varepsilon^{x^{2}}-1 \sim x^{2}, \text{ as } x^{x} \rightarrow 0\left(x^{2} \rightarrow 0\right)
\end{aligned}
$$

We get:

$$
\lim _{x \rightarrow \overline{\overline{0}}} \frac{1-\cos 10 x}{e^{x^{2}}-1}=\left\{\frac{0}{0}\right\}=\lim _{x \rightarrow \overline{0}} \frac{\frac{100 x^{2}}{2}}{x^{2}}=\lim _{x \rightarrow \overline{0}} \frac{100}{2}=50
$$

## Problem Kuznetsov Limits 12-2