## Problem Statement

Calculate the limit of the function:

$$
\lim _{x \rightarrow 0} \frac{(1+x)^{3}-(1+3 x)}{x+x^{5}}
$$
## Solution

$$
\begin{aligned}
& \lim _{x \rightarrow 0} \frac{(1+x)^{3}-(1+3 x)}{x+x^{5}}=\left\{\frac{0}{0}\right\}=\lim _{x \rightarrow 0} \frac{1^{: 5}+3 \cdot 1^{2} \cdot x^{2}+3 \cdot 1 \cdot x^{2}+x^{3}-1-3 x}{x\left(1+x^{4}\right)}= \\
& =\lim _{x \rightarrow 0} \frac{1+3 x+3 x^{2}+x^{3}-1-3 x}{x\left(1+x^{4}\right)}=\lim _{x \rightarrow 0} \frac{3 x^{2}+x^{3}}{x\left(1+x^{4}\right)}= \\
& =\lim _{x \rightarrow 0} \frac{x^{2}(3+x)}{x\left(1+x^{4}\right)}=\lim _{x \rightarrow 0} \frac{x\left(3+x^{3}\right)}{1+x^{4}}=\frac{0(3+0)}{1+0^{4}}=\frac{0}{1}=0
\end{aligned}
$$

## Problem Kuznetsov Limits 10-7