## Problem Statement

Calculate the limit of the function:

$$
\lim _{x \rightarrow 0}\left(1+\tan^{2} x\right)^{\frac{1}{\ln \left(1+3 x^{2}\right)}}
$$
## Solution

$\lim _{x \rightarrow 0}\left(1+\tan ^{2} x\right)^{\frac{1}{\ln \left(1+3 x^{2}\right)}}=$

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

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

$=\exp \left\{\lim _{x \rightarrow 0} \frac{\ln \left(1+\tan ^{2} x\right)}{\ln \left(1+3 x^{2}\right)}\right\}=$

Using the substitution of equivalent infinitesimals:

$$
\begin{aligned}
& \ln \left(1+\tan ^{2} x\right) \sim \tan ^{2} x, \text { as } x \rightarrow 0\left(\tan ^{2} x \rightarrow 0\right) \\
& \ln \left(1+3 x^{2}\right) \sim 3 x^{2}, \text { as } x \rightarrow 0\left(3 x^{2} \rightarrow 0\right)
\end{aligned}
$$

We get:

$$
=\exp \left\{\lim _{x \rightarrow 0} \frac{\tan ^{2} x}{3 x^{2}}\right\}=
$$

Using the substitution of equivalent infinitesimals:

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

We get:

$=\exp \left\{\lim _{x \rightarrow 0} \frac{x^{2}}{3 x^{2}}\right\}=\exp \left\{\lim _{x \rightarrow 0} \frac{1}{3}\right\}=e^{\frac{1}{3}}$

## Problem Kuznetsov Limits 17-28