## Problem Statement

Calculate the limit of the function:

$\lim _{x \rightarrow 0} \frac{\sin ^{2} x-\tan^{2} x}{x^{4}}$
## Solution

We will use the substitution of equivalent infinitesimals:

$$
\begin{aligned}
& \sin x \sim x, \text { as } x \rightarrow 0 \\
& \operatorname{tg} x \sim x, \text { as } x \rightarrow 0
\end{aligned}
$$

We get:

$$
\begin{aligned}
& \lim _{x \rightarrow 0} \frac{\sin ^{2} x-\operatorname{tg}^{2} x}{x^{4}}=\left\{\frac{0}{0}\right\}=\lim _{x \rightarrow 0} \frac{\frac{\sin ^{2} x \cdot \cos ^{2} x}{\cos ^{2} x}-\operatorname{tg}^{2} x}{x^{4}}= \\
& =\lim _{x \rightarrow 0} \frac{\operatorname{tg}^{2} x \cdot \cos ^{2} x-\operatorname{tg}^{2} x}{x^{4}}=\lim _{x \rightarrow 0} \frac{\operatorname{tg}^{2} x\left(\cos ^{2} x-1\right)}{x^{4}}=
\end{aligned}
$$

$$
=\lim _{x \rightarrow 0} \frac{x^{2}\left(-\sin ^{2} x\right)}{x^{4}}=\lim _{x \rightarrow 0} \frac{x^{2}\left(-x^{2}\right)}{x^{4}}=\lim _{x \rightarrow 0} \frac{-1}{1}=-1
$$

## Problem Kuznetsov Limits 12-25