Condition of the problem

Calculate the limit of the function:
$\lim _{x \rightarrow 0} \frac{(1+x)^{3}-(1+3 x)}{x^{2}+x^{5}}$
## Solution

$\lim _{x \rightarrow 0} \frac{(1+x)^{3}-(1+3 x)}{x^{2}+x^{5}}=\left\{\frac{0}{0}\right\}=\lim _{x \rightarrow 0} \frac{1^{3}+3 \cdot 1^{2} \cdot x+3 \cdot 1 \cdot x^{2}+x^{3}-1-3 x}{x^{2}\left(1+x^{3}\right)}=$
$=\lim _{x \rightarrow 0} \frac{1+3 x+3 x^{2}+x^{3}-1-3 x}{x^{2}\left(1+x^{3}\right)}=\lim _{x \rightarrow 0} \frac{3 x^{2}+x^{3}}{x^{2}\left(1+x^{3}\right)}=$
$=\lim _{x \rightarrow 0} \frac{x^{2}(3+x)}{x^{2}\left(1+x^{3}\right)}=\lim _{x \rightarrow 0} \frac{3+x}{1+x^{3}}=\frac{3+0}{1+0^{3}}=\frac{3}{1}=3$

## Problem Kuznetsov Limits 10-28