## Problem Statement

Calculate the limit of the function:

$$
\lim _{x \rightarrow 0}(1-\ln (1+\sqrt[3]{x}))^{\frac{x}{\sin ^{4} \sqrt[3]{x}}}
$$
## Solution

$$
\begin{aligned}
& \lim _{x \rightarrow 0}(1-\ln (1+\sqrt[3]{x}))^{\frac{\sin ^{4} \sqrt[3]{x}}{x}}= \\
& =\lim _{x \rightarrow 0}\left(e^{\ln (1-\ln (1+\sqrt[3]{x})))^{\frac{x}{\sin ^{4} \sqrt[3]{x}}}=}\right. \\
& =\lim _{x \rightarrow 0} e^{\frac{x \ln (1-\ln (1+\sqrt[3]{x}))}{\sin ^{4} \sqrt[3]{x}}}= \\
& =\exp \left\{\lim _{x \rightarrow 0} \frac{x \ln (1-\ln (1+\sqrt[3]{x}))}{\sin ^{4} \sqrt[3]{x}}\right\}=
\end{aligned}
$$

Using the substitution of equivalent infinitesimals:

$$
\begin{aligned}
& \ln (1-\ln (1+\sqrt[3]{x})) \sim-\ln (1+\sqrt[3]{x}), \text { as } x \rightarrow 0(-\ln (1+\sqrt[3]{x}) \rightarrow 0) \\
& \sin \sqrt[3]{x} \sim \sqrt[3]{x}, \text { as } \\
& x \rightarrow 0(\sqrt[3]{x} \rightarrow 0)
\end{aligned}
$$

We get:

$=\exp \left\{\lim _{x \rightarrow 0} \frac{x(-\ln (1+\sqrt[3]{x}))}{(\sqrt[3]{x})^{4}}\right\}=$

Using the substitution of equivalent infinitesimals:

$\ln (1+\sqrt[3]{x}) \sim \sqrt[3]{x}$, as $x \rightarrow 0(\sqrt[3]{x} \rightarrow 0)$

We get:

$=\exp \left\{\lim _{x \rightarrow 0} \frac{x(-\sqrt[3]{x})}{x \cdot \sqrt[3]{x}}\right\}=\exp \left\{\lim _{x \rightarrow 0} \frac{-1}{1}\right\}=e^{-1}$

## Problem Kuznetsov Limits 17-7